Unit: 13: Exponents and Powers

Exercise: 1 (Multiple Choice Questions and Answers 1-22)

In questions 1 to 22, there are four options, out of which one is correct. Write the correct one.

Question: 1

[ ( 3 ) 2 ] 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada qadaqaaiabgkHiTiaaiodaaiaawIcacaGLPaaadaahaaWcbeqaaiaa ikdaaaaakiaawUfacaGLDbaadaahaaWcbeqaaiaaiodaaaaaaa@3CDB@  is equal to

a.    ( 3 ) 8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq GHsislcaaIZaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI4aaaaaaa @39FB@

b.   ( 3 ) 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq GHsislcaaIZaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI2aaaaaaa @39F9@

c.    ( 3 ) 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq GHsislcaaIZaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI1aaaaaaa @39F8@

d.   ( 3 ) 23 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq GHsislcaaIZaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaGaaG4m aaaaaaa@3AB2@

Solution

(b)

We have studied that, if a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiigGiaadg gacaGGzacaaa@3837@  is a rational number, m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGTbaaaa@36EA@  and n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36CC@   are natural numbers, then ( a m ) n  = a m×n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadg gadaahaaWcbeqaaiaad2gaaaGccaGGPaWaaWbaaSqabeaacaWGUbaa aOGaaiiOaiabg2da9iaadggadaahaaWcbeqaaiaad2gacqGHxdaTca WGUbaaaaaa@41A3@   a=(3),m=2,n=3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2 da9iaacIcacqGHsislcaaIZaGaaiykaiaacYcacaWGTbGaeyypa0Ja aGOmaiaacYcacaWGUbGaeyypa0JaaG4maaaa@4191@

So,  [ (3) 2 ] 3  = ( 3 ) 2×3  = ( 3 ) 6 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4uaiaab+ gacaqGSaGaaeiiaiaacUfacaGGOaGaeyOeI0IaaG4maiaacMcadaah aaWcbeqaaiaaikdaaaGccaGGDbWaaWbaaSqabeaacaaIZaaaaOGaai iOaiabg2da9maabmaabaGaeyOeI0IaaG4maaGaayjkaiaawMcaamaa CaaaleqabaGaaGOmaiabgEna0kaaiodaaaGccaGGGcGaeyypa0Zaae WaaeaacqGHsislcaaIZaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI 2aaaaOGaaiOlaaaa@4FC6@

Question: 2

For a non-zero rational number x, x 8 ÷ x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaacY cacaaMc8UaaGPaVlaadIhadaahaaWcbeqaaiaaiIdaaaGccqGH3daU caWG4bWaaWbaaSqabeaacaaIYaaaaaaa@40B3@  is equal to

a.    x 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGinaaaaaaa@37C1@

b.   x 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGOnaaaaaaa@37C3@

c.    x 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGymaiaaicdaaaaaaa@3878@

d.   x 16 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGymaiaaiAdaaaaaaa@387E@

Solution

(b)

We have studied that, if a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiigGiaadg gacaGGzacaaa@3837@  is a rational number, m and n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiaabc cacaqGHbGaaeOBaiaabsgacaqGGaGaamOBaaaa@3BBF@  are natural numbers such that m>n, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabg6 da+iaad6gacaGGSaaaaa@3975@  then a m ÷ a n = a (mn) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaamyBaaaakiabgEpa4kaadggadaahaaWcbeqaaiaad6ga aaGccqGH9aqpcaWGHbWaaWbaaSqabeaacaGGOaGaamyBaiabgkHiTi aad6gacaGGPaaaaaaa@4276@

Here, we have a=x, m=8, n=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2 da9iaadIhacaGGSaGaaeiiaiaad2gacqGH9aqpcaaI4aGaaiilaiaa bccacaWGUbGaeyypa0JaaGOmaaaa@40D6@

So, x 8 ÷ x 2 = x 8 x 2 = x 82 = x 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabaeaacaWG4b WaaWbaaSqabeaacaaI4aaaaOGaey49aGRaamiEamaaCaaaleqabaGa aGOmaaaakiabg2da9maalaaabaGaamiEamaaCaaaleqabaGaaGioaa aaaOqaaiaadIhadaahaaWcbeqaaiaaikdaaaaaaaGcbaGaeyypa0Ja amiEamaaCaaaleqabaGaaGioaiabgkHiTiaaikdaaaaakeaacqGH9a qpcaWG4bWaaWbaaSqabeaacaaI2aaaaaaaaa@4894@  

Question: 3

x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D6@  is a non-zero rational number. Product of the square of x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D6@  with the cube of x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D6@  is equal to the

a.    second power of x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D6@

b.   third power of x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D6@

c.    fifth power of x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D6@

d.   sixth power of x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D6@

Solution

(c)

Square of x= x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 da9iaadIhadaahaaWcbeqaaiaaikdaaaaaaa@39C2@

Cube of x= x 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 da9iaadIhadaahaaWcbeqaaiaaiodaaaaaaa@39C2@

Product of square with the cube of x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D6@    = x 2 ×  x 3 = x 2+3 [ a m × a n = a m+n ] = x 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey41aqRaaeiiaiaadIha daahaaWcbeqaaiacmciIZaaaaaGcbaGaeyypa0JaamiEamaaCaaale qabaGaaGOmaiabgUcaRiaaiodaaaGccaaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caGGBbGaeyinIWLaamyyamaaCaaaleqabaGaam yBaaaakiabgEna0kaadggadaahaaWcbeqaaiaad6gaaaGccqGH9aqp caWGHbWaaWbaaSqabeaacaWGTbGaey4kaSIaamOBaaaakiaac2faae aacqGH9aqpcaWG4bWaaWbaaSqabeaacaaI1aaaaaaaaa@5D79@

i.e. fifth power of x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D6@ .

Question: 4

For any two non-zero rational numbers x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D6@  and y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36D7@ , x 5 ÷ y 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGynaaaakiabgEpa4kaadMhadaahaaWcbeqaaiaaiwda aaaaaa@3BF1@  is equal to

a.    ( x÷y ) 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WG4bGaey49aGRaamyEaaGaayjkaiaawMcaamaaCaaaleqabaGaaGym aaaaaaa@3C80@

b.   ( x÷y ) 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WG4bGaey49aGRaamyEaaGaayjkaiaawMcaamaaCaaaleqabaGaaGim aaaaaaa@3C7F@

c.    ( x÷y ) 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WG4bGaey49aGRaamyEaaGaayjkaiaawMcaamaaCaaaleqabaGaaGyn aaaaaaa@3C84@

d.   ( x÷y ) 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WG4bGaey49aGRaamyEaaGaayjkaiaawMcaamaaCaaaleqabaGaaGym aiaaicdaaaaaaa@3D3A@

Solution

(c)

It is given that, x 5 ÷ y 5 = x 5 y 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGynaaaakiabgEpa4kaadMhadaahaaWcbeqaaiaaiwda aaGccqGH9aqpdaWcaaqaaiaadIhadaahaaWcbeqaaiaaiwdaaaaake aacaWG5bWaaWbaaSqabeaacaaI1aaaaaaaaaa@40ED@

We have studied that, p n q n = ( p q ) n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGWbWaaWbaaSqabeaacaWGUbaaaaGcbaGaamyCamaaCaaaleqabaGa amOBaaaaaaGccqGH9aqpdaqadaqaamaalaaabaGaamiCaaqaaiaadg haaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaWGUbaaaaaa@3FD1@

Here, p=x, q=y, n=5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2 da9iaadIhacaGGSaGaaeiiaiaadghacqGH9aqpcaWG5bGaaiilaiaa bccacaWGUbGaeyypa0JaaGynaaaa@4128@

Thus, x 5 y 5 = ( x y ) 5 = (x÷y) 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabaeaadaWcaa qaaiaadIhadaahaaWcbeqaaiaaiwdaaaaakeaacaWG5bWaaWbaaSqa beaacaaI1aaaaaaakiabg2da9maabmaabaWaaSaaaeaacaWG4baaba GaamyEaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiacmciI1aaaaaGc baGaeyypa0JaaiikaiaadIhacqGH3daUcaWG5bGaaiykamaaCaaale qabaGaaGynaaaaaaaa@47FD@

Question: 5

a m × a n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaamyBaaaakiabgEna0kaadggadaahaaWcbeqaaiaad6ga aaaaaa@3C05@  is equal to

a.    ( a 2 ) mn MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadg gadaahaaWcbeqaaiaaikdaaaGccaGGPaWaaWbaaSqabeaacaWGTbGa amOBaaaaaaa@3B1D@

b.   a mn MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaamyBaiabgkHiTiaad6gaaaaaaa@39BE@

c.    a m+n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaamyBaiabgUcaRiaad6gaaaaaaa@39B3@

d.   a mn MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaamyBaiaad6gaaaaaaa@38D1@

Solution

(c)

As we have studied that, if a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiigGiaadg gacaGGzacaaa@3837@  is a rational number, m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@36CA@  and n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36CC@  are natural numbers, then a m × a n = a m+n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaamyBaaaakiabgEna0kaadggadaahaaWcbeqaaiaad6ga aaGccqGH9aqpcaWGHbWaaWbaaSqabeaacaWGTbGaey4kaSIaamOBaa aaaaa@40EE@  

Question: 6

( 1 0 + 2 0 + 3 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIXaWaaWbaaSqabeaacaaIWaaaaGGaaOGae83kaSIae8NmaiZaaWba aSqabeaacqWFWaamaaGccqWFRaWkcqWFZaWmdaahaaWcbeqaaiab=b daWaaaaOGaayjkaiaawMcaaaaa@3EDA@  is equal to

a.    0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@3693@

b.   1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaaaa@3694@

c.    3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maaaa@3696@

d.   6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnaaaa@3699@

Solution

(c)

As we have studied that, a 0 =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaGimaaaakiabg2da9iaaigdaaaa@3970@

MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaGaeyinIWfaaa@3715@   1 0 + 2 0 + 3 0 =1+1+1 =3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaaIXa WaaWbaaSqabeaacaaIWaaaaOGaey4kaSIaaGOmamaaCaaaleqabaGa aGimaaaakiabgUcaRiaaiodadaahaaWcbeqaaiaaicdaaaaakeaacq GH9aqpcaaIXaGaey4kaSIaaGymaiabgUcaRiaaigdaaeaacqGH9aqp caaIZaaaaaa@4368@

Question: 7

Value of 10 22 + 10 20 10 20 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaGaaGimamaaCaaaleqabaGaaGOmaiaaikdaaaGccqGHRaWkcaaI XaGaaGimamaaCaaaleqabaGaaGOmaiaaicdaaaaakeaacaaIXaGaaG imamaaCaaaleqabaGaaGOmaiaaicdaaaaaaaaa@4029@  is

a.    10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic daaaa@374E@

b.   10 42 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic dadaahaaWcbeqaaiaaisdacaaIYaaaaaaa@38F5@

c.    101 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic dacaaIXaaaaa@3809@

d.   10 22 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic dadaahaaWcbeqaaiaaikdacaaIYaaaaaaa@38F3@

Solution

(c)

 The given expression can be written as

  10 22 10 20 + 10 20 10 20 = 10 2220 + 10 2020 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWcaa qaaiaaigdacaaIWaWaaWbaaSqabeaacaaIYaGaaGOmaaaaaOqaaiaa igdacaaIWaWaaWbaaSqabeaacaaIYaGaaGimaaaaaaGccqGHRaWkda WcaaqaaiaaigdacaaIWaWaaWbaaSqabeaacaaIYaGaaGimaaaaaOqa aiaaigdacaaIWaWaaWbaaSqabeaacaaIYaGaaGimaaaaaaaakeaacq GH9aqpcaaIXaGaaGimamaaCaaaleqabaGaaGOmaiaaikdacqGHsisl caaIYaGaaGimaaaakiabgUcaRiaaigdacaaIWaWaaWbaaSqabeaaca aIYaGaaGimaiabgkHiTiaaikdacaaIWaaaaaaaaa@5055@

= 10 2 +1 =10×10+1 =100+1 =101 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaaIXaGaaGimamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaigda aeaacqGH9aqpcaaIXaGaaGimaiabgEna0kaaigdacaaIWaGaey4kaS IaaGymaaqaaiabg2da9iaaigdacaaIWaGaaGimaiabgUcaRiaaigda aeaacqGH9aqpcaaIXaGaaGimaiaaigdaaaaa@4A98@

Question: 8

The standard form of the number 12345 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaik dacaaIZaGaaGinaiaaiwdaaaa@398A@  is

a.    12345× 10 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaik dacaaIZaGaaGinaiaaiwdacqGHxdaTcaaIXaGaaGimamaaCaaaleqa baGaaGymaaaaaaa@3DFE@

b.   123.45× 10 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaik dacaaIZaGaaiOlaiaaisdacaaI1aGaey41aqRaaGymaiaaicdadaah aaWcbeqaaiaaikdaaaaaaa@3EB1@

c.    12.345× 10 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaik dacaGGUaGaaG4maiaaisdacaaI1aGaey41aqRaaGymaiaaicdadaah aaWcbeqaaiaaiodaaaaaaa@3EB2@

d.   1.2345× 10 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaac6 cacaaIYaGaaG4maiaaisdacaaI1aGaey41aqRaaGymaiaaicdadaah aaWcbeqaaiaaisdaaaaaaa@3EB3@

Solution

(d)

A number in its standard form is written as ‘a × 10 k , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey41aqRaaG jbVlaaigdacaaIWaWaaWbaaSqabeaacaWGRbaaaOGaaiilaaaa@3CC8@  where a is a terminating decimal number such that 1a<10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgs MiJkaadggacqGH8aapcaaIXaGaaGimaaaa@3BA7@  and k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@36C9@  is any integer.

Thus, the standard form of the number 12345=1.2345× 10 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaik dacaaIZaGaaGinaiaaiwdacqGH9aqpcaaIXaGaaiOlaiaaikdacaaI ZaGaaGinaiaaiwdacqGHxdaTcaaIXaGaaGimamaaCaaaleqabaGaaG inaaaaaaa@4369@

Question: 9

If 2 1998 2 1997 2 1996 2 1995 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaGymaiaaiMdacaaI5aGaaGioaaaakiabgkHiTiaaikda daahaaWcbeqaaiaaigdacaaI5aGaaGyoaiaaiEdaaaGccqGHsislca aIYaWaaWbaaSqabeaacaaIXaGaaGyoaiaaiMdacaaI2aaaaOGaeyOe I0IaaGOmamaaCaaaleqabaGaaGymaiaaiMdacaaI5aGaaGynaaaaaa a@4868@   =K. 2 1995 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaae 4saiaab6cacaaMc8UaaGOmamaaCaaaleqabaGaaGymaiaaiMdacaaI 5aGaaGynaaaakiaaykW7aaa@3F67@ , then the value of K is

a.    1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaaaa@3694@

b.   2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaaaa@3695@

c.    3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maaaa@3696@

d.   4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaaaa@3697@

Solution

(c) It is given that

2 1998 2 1997 2 1996 + 2 1995 =K .2 1995 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaGymaiaaiMdacaaI5aGaaGioaaaakiaacobicaaIYaWa aWbaaSqabeaacaaIXaGaaGyoaiaaiMdacaaI3aaaaOGaai4eGiaaik dadaahaaWcbeqaaiaaigdacaaI5aGaaGyoaiaaiAdaaaGccqGHRaWk caaIYaWaaWbaaSqabeaacaaIXaGaaGyoaiaaiMdacaaI1aaaaOGaey ypa0Jaae4saiaac6cacaaIYaWaaWbaaSqabeaacaaIXaGaaGyoaiaa iMdacaaI1aaaaaaa@4E69@

2 1995 +3 2 1995+2 2 1995+1 + 2 1995 =K .2 1995 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4TaaG OmamaaCaaaleqabaGaaGymaiaaiMdacaaI5aGaaGynaaaakmaaCaaa leqabaGaey4kaSIaaG4maaaakiabgkHiTiaaikdadaahaaWcbeqaai aaigdacaaI5aGaaGyoaiaaiwdacqGHRaWkcaaIYaaaaOGaeyOeI0Ia aGOmamaaCaaaleqabaGaaGymaiaaiMdacaaI5aGaaGynaiabgUcaRi aaigdaaaGccqGHRaWkcaaIYaWaaWbaaSqabeaacaaIXaGaaGyoaiaa iMdacaaI1aaaaOGaeyypa0Jaae4saiaac6cacaaIYaWaaWbaaSqabe aacaaIXaGaaGyoaiaaiMdacaaI1aaaaaaa@563D@

2 1995 [ 2 3 2 2 2 1 +1]=K .2 1995 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taiq hGikdadGaDaYbaaSqajqhGbGaDakac0biIXaGaiqhGiMdacGaDaIyo aiac0biI1aaaaOGaiqhGcUfacGaDaIOmamac0bihaaWcbKaDagac0b OaiqhGiodaaaGccWaDaAOeI0IaiqhGikdadGaDaYbaaSqajqhGbGaD akac0biIYaaaaOGamqhGgkHiTiac0biIYaWaiqhGCaaaleqc0byaiq hGcGaDaIymaaaakiad0bOHRaWkcGaDaIymaiac0bOGDbGamqhGg2da 9iaabUeacGaDakOlaiac0biIYaWaiqhGCaaaleqc0byaiqhGcGaDaI ymaiac0biI5aGaiqhGiMdacGaDaIynaaaaaaa@733F@

2 1995 [842+1]=K .2 1995 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4TaaG OmamaaCaaaleqabaGaaGymaiaaiMdacaaI5aGaaGynaaaakiaacUfa caaI4aGaeyOeI0IaaGinaiabgkHiTiaaikdacqGHRaWkcaaIXaGaai yxaiabg2da9iaabUeacaGGUaGaaGOmamaaCaaaleqabaGaaGymaiaa iMdacaaI5aGaaGynaaaaaaa@4A0A@

3= K .2 1995 2 1995 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4TaaG 4maiabg2da9maalaaabaGaae4saiaac6cacaaIYaWaaWbaaSqabeaa caaIXaGaaGyoaiaaiMdacaaI1aaaaaGcbaGaaGOmamaaCaaaleqaba GaaGymaiaaiMdacaaI5aGaaGynaaaaaaaaaa@4364@

3=K or K=3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4TaaG 4maiabg2da9iaabUeacaqGGaGaae4BaiaabkhacaqGGaGaae4saiab g2da9iaaiodaaaa@4084@

Thus, the value of K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4saaaa@36A7@  is 3. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaac6 caaaa@3748@

Question:10

Which of the following is equal to 1?

a.    2 0 + 3 0 + 4 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaaccaGae8hmaadaaOGae83kaSIae83mamZaaWbaaSqabeaa cqWFWaamaaGccqGHRaWkcaaI0aWaaWbaaSqabeaacqWFWaamaaaaaa@3D4E@

b.   2 0 × 3 0 × 4 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaaccaGae8hmaadaaOGaey41aqRaaG4mamaaCaaaleqabaGa e8hmaadaaOGaey41aqRaaGinamaaCaaaleqabaGaaGimaaaaaaa@3F66@

c.    ( 3 0 2 0 )× 4 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIZaWaaWbaaSqabeaacaaIWaaaaOGaeyOeI0IaaGOmamaaCaaaleqa baaccaGae8hmaadaaaGccaGLOaGaayzkaaGaey41aqRaaGinamaaCa aaleqabaGae8hmaadaaaaa@3FC5@

d.   ( 3 0 2 0 )×( 3 0 + 2 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIZaWaaWbaaSqabeaaiiaacqWFWaamaaGccqGHsislcaaIYaWaaWba aSqabeaacqWFWaamaaaakiaawIcacaGLPaaacqGHxdaTdaqadaqaai aaiodadaahaaWcbeqaaiab=bdaWaaakiabgUcaRiaaikdadaahaaWc beqaaiab=bdaWaaaaOGaayjkaiaawMcaaaaa@443D@

Solution

(b)

Let us solve all the expressions one by one,

Option (a),

2 0 + 3 0 + 4 0 =1+1+1 =3      [ a 0 =1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaGimaaaakiabgUcaRiaaiodadaahaaWcbeqaaiaaicda aaGccqGHRaWkcaaI0aWaaWbaaSqabeaacaaIWaaaaOGaeyypa0JaaG ymaiabgUcaRiaaigdacqGHRaWkcaaIXaGaaiiOaiabg2da9iaaioda caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccadaWadaqaaiablw JirjaadggadaahaaWcbeqaaiaaicdaaaGccqGH9aqpcaaIXaaacaGL BbGaayzxaaaaaa@4F1D@

Option (b),

2 0 × 3 0 × 4 0 =1×1×1 =1         [ a 0 =1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaGimaaaakiabgEna0kaaiodadaahaaWcbeqaaiaaicda aaGccqGHxdaTcaaI0aWaaWbaaSqabeaacaaIWaaaaOGaeyypa0JaaG ymaiabgEna0kaaigdacqGHxdaTcaaIXaGaaiiOaiabg2da9iaaigda caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccadaWadaqaaiablwJirjaadggadaahaaWcbeqaaiaaicdaaaGc cqGH9aqpcaaIXaaacaGLBbGaayzxaaaaaa@55D8@

Hence, option (b) is the answer.

Question: 11

In standard form, the number 72105.4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4naiaaik dacaaIXaGaaGimaiaaiwdacaGGUaGaaGinaaaa@3AFA@  is written as 7.21054× 10 n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4naiaac6 cacaaIYaGaaGymaiaaicdacaaI1aGaaGinaiabgEna0kaaigdacaaI WaWaaWbaaSqabeaacaWGUbaaaaaa@3FA6@  where n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36CC@  is equal to

a.    2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaaaa@3695@

b.   3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maaaa@3696@

c.    4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaaaa@3697@

d.   5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaaaa@3698@

Solution

(c)

We have studied that, if the given number is greater than or equal to 10, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic dacaGGSaaaaa@37FE@  then the power of 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic daaaa@374E@  (i.e. n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36CC@  ) is a positive integer and is equal to the number of places the decimal point has been shifted.

Hence, 72105.4=7.21054× 10 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4naiaaik dacaaIXaGaaGimaiaaiwdacaGGUaGaaGinaiabg2da9iaaiEdacaGG UaGaaGOmaiaaigdacaaIWaGaaGynaiaaisdacqGHxdaTcaaIXaGaaG imamaaCaaaleqabaGaaGinaaaaaaa@4597@    

Question: 12

Square of ( 2 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiabgkHiTiaaikdaaeaacaaIZaaaaaGaayjkaiaawMcaaaaa @39D8@  is

a.    2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIYaaabaGaaG4maaaaaaa@384F@

b.   2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIYaaabaGaaG4maaaaaaa@3762@

c.    4 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI0aaabaGaaGyoaaaaaaa@3857@

d.   4 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI0aaabaGaaGyoaaaaaaa@376A@

Solution

(d)

As per question, square of 2 3   is  ( 2 3 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaS aaaeaacaaIYaaabaGaaG4maaaacaGGGcGaaeiiaiaabMgacaqGZbGa aeiiamaabmaabaGaeyOeI0YaaSaaaeaacaaIYaaabaGaaG4maaaaai aawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaaaa@4182@

Thus, ( 2 3 ) 2                                        ( 1 ) 2 =1 =( 2 3 )×( 2 3 ) = 4 9        MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaqada qaaiabgkHiTmaalaaabaGaaGOmaaqaaiaaiodaaaaacaGLOaGaayzk aaWaaWbaaSqabeaacaaIYaaaaOGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacqWI1isudaqadaqaaiabgkHiTiaaigdaaiaawIcacaGL PaaadaahaaWcbeqaaiaaikdaaaGccqGH9aqpcaaIXaaabaGaeyypa0 ZaaeWaaeaacqGHsisldaWcaaqaaiaaikdaaeaacaaIZaaaaaGaayjk aiaawMcaaiabgEna0oaabmaabaGaeyOeI0YaaSaaaeaacaaIYaaaba GaaG4maaaaaiaawIcacaGLPaaaaeaacqGH9aqpdaWcaaqaaiaaisda aeaacaaI5aaaaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaa aaaa@6BA5@

Question: 13

Cube of ( 1 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiabgkHiTiaaigdaaeaacaaI0aaaaaGaayjkaiaawMcaaaaa @39D8@  is

a.    1 12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaS aaaeaacaaIXaaabaGaaGymaiaaikdaaaaaaa@3908@

b.   1 16 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGymaiaaiAdaaaaaaa@381F@

c.    1 64 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaS aaaeaacaaIXaaabaGaaGOnaiaaisdaaaaaaa@390F@

d.   1 64 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGOnaiaaisdaaaaaaa@3822@

Solution

(c)

As per question, cube of ( 1 4 ) is  ( 1 4 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabgk HiTmaalaaabaGaaGymaaqaaiaaisdaaaGaaiykaiaabccacaqGPbGa ae4CaiaabccacaGGOaGaeyOeI0YaaSaaaeaacaaIXaaabaGaaGinaa aacaGGPaWaaWbaaSqabeaacaaIZaaaaaaa@4188@

Thus, ( 1 4 ) 3 =( 1 4 )×( 1 4 )×( 1 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq GHsisldaWcaaqaaiaabgdaaeaacaaI0aaaaaGaayjkaiaawMcaamaa CaaaleqabaGaae4maaaakiabg2da9maabmaabaGaeyOeI0YaaSaaae aacaqGXaaabaGaaGinaaaaaiaawIcacaGLPaaacqGHxdaTdaqadaqa aiabgkHiTmaalaaabaGaaeymaaqaaiaaisdaaaaacaGLOaGaayzkaa Gaey41aq7aaeWaaeaacqGHsisldaWcaaqaaiaabgdaaeaacaaI0aaa aaGaayjkaiaawMcaaaaa@4BD9@

= (1)×(1)×(1) 4×4×4                     ( 1 ) 3 =1 = 1 64 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpdaWcaaqaaiaacIcacqGHsislcaqGXaGaaiykaiabgEna0kaacIca cqGHsislcaqGXaGaaiykaiabgEna0kaacIcacqGHsislcaqGXaGaai ykaaqaaiaaisdacqGHxdaTcaaI0aGaey41aqRaaGinaaaacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaeSynIe1aaeWaaeaacqGHsislcaaIXaaacaGLOaGaay zkaaWaaWbaaSqabeaacaaIZaaaaOGaeyypa0JaeyOeI0IaaGymaaqa aiabg2da9iabgkHiTmaalaaabaGaaeymaaqaaiaaiAdacaaI0aaaaa aaaa@62D3@

Question: 14

Which of the following is not equal to ( 5 4 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiabgkHiTiaaiwdaaeaacaaI0aaaaaGaayjkaiaawMcaamaa CaaaleqabaGaaGinaaaaaaa@3AC7@ ?

a.    MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcaa@35D8@ ( 5 ) 4 4 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaada qadaqaaiabgkHiTiaaiwdaaiaawIcacaGLPaaadaahaaWcbeqaaiaa isdaaaaakeaacaaI0aWaaWbaaSqabeaacaaI0aaaaaaaaaa@3BBC@

b.   5 4 ( 4 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI1aWaaWbaaSqabeaacaaI0aaaaaGcbaWaaeWaaeaacqGHsislcaaI 0aaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI0aaaaaaaaaa@3BBC@

c.    5 4 4 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaS aaaeaacaaI1aWaaWbaaSqabeaacaaI0aaaaaGcbaGaaGinamaaCaaa leqabaGaaGinaaaaaaaaaa@3A33@

d.   ( 5 4 )×( 5 4 )×( 5 4 )×( 5 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq GHsisldaWcaaqaaiaaiwdaaeaacaaI0aaaaaGaayjkaiaawMcaaiab gEna0oaabmaabaGaeyOeI0YaaSaaaeaacaaI1aaabaGaaGinaaaaai aawIcacaGLPaaacqGHxdaTdaqadaqaaiabgkHiTmaalaaabaGaaGyn aaqaaiaaisdaaaaacaGLOaGaayzkaaGaey41aq7aaeWaaeaacqGHsi sldaWcaaqaaiaaiwdaaeaacaaI0aaaaaGaayjkaiaawMcaaaaa@4C2A@

Solution

(c)

We have studied that, ( p q ) m = p m q m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaadchaaeaacaWGXbaaaaGaayjkaiaawMcaamaaCaaaleqa baGaiWiGd2gaaaGccqGH9aqpdaWcaaqaaiaadchadaahaaWcbeqaai aad2gaaaaakeaacaWGXbWaaWbaaSqabeaacaWGTbaaaaaaaaa@40E2@

So, ( 5 4 ) 4 = (5) 4 4 4   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq GHsisldaWcaaqaaiaaiwdaaeaacaaI0aaaaaGaayjkaiaawMcaamaa CaaaleqabaGaaGinaaaakiabg2da9maalaaabaGaaiikaiabgkHiTi aaiwdacaGGPaWaaWbaaSqabeaacaaI0aaaaaGcbaGaaGinamaaCaaa leqabaGaaGinaaaaaaGccaqGGaaaaa@4236@

or  ( 5 4 ) 4 = (5) 4 (4) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4Baiaabk hacaqGGaWaaeWaaeaacqGHsisldaWcaaqaaiaaiwdaaeaacaaI0aaa aaGaayjkaiaawMcaamaaCaaaleqabaGaaGinaaaakiabg2da9maala aabaGaaiikaiaaiwdacaGGPaWaaWbaaSqabeaacaaI0aaaaaGcbaGa aiikaiabgkHiTiaaisdacaGGPaWaaWbaaSqabeaacaaI0aaaaaaaaa a@456C@

Or  ( 5 4 ) 4 =( 5 4 )×( 5 4 )×( 5 4 )×( 5 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaqada qaaiabgkHiTmaalaaabaGaaGynaaqaaiaaisdaaaaacaGLOaGaayzk aaWaaWbaaSqabeaacaqG0aaaaaGcbaGaeyypa0ZaaeWaaeaacqGHsi sldaWcaaqaaiaaiwdaaeaacaaI0aaaaaGaayjkaiaawMcaaiabgEna 0oaabmaabaGaeyOeI0YaaSaaaeaacaaI1aaabaGaaGinaaaaaiaawI cacaGLPaaacqGHxdaTdaqadaqaaiabgkHiTmaalaaabaGaaeynaaqa aiaaisdaaaaacaGLOaGaayzkaaGaey41aq7aaeWaaeaacqGHsislda WcaaqaaiaaiwdaaeaacaaI0aaaaaGaayjkaiaawMcaaaaaaa@521F@

Hence, option (c) is not equal to 1. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaac6 caaaa@3746@

Question: 15

Which of the following is not equal to 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaaaa@3694@ ?

a.    2 3 × 3 2 4×18 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIYaWaaWbaaSqabeaacaaIZaaaaOGaey41aqRaaG4mamaaCaaaleqa baGaaGOmaaaaaOqaaiaaisdacqGHxdaTcaaIXaGaaGioaaaaaaa@3FB1@

b.   [ ( 2 ) 3 × ( 2 ) 4 ]÷ ( 2 ) 7 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada qadaqaaiabgkHiTiaaikdaaiaawIcacaGLPaaadaahaaWcbeqaaiaa iodaaaGccqGHxdaTdaqadaqaaiabgkHiTiaaikdaaiaawIcacaGLPa aadaahaaWcbeqaaiaaisdaaaaakiaawUfacaGLDbaacqGH3daUdaqa daqaaiabgkHiTiaaikdaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiE daaaaaaa@488A@

c.    3 0 × 5 3 5×25 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaWaaWbaaSqabeaacaaIWaaaaOGaey41aqRaaGynamaaCaaaleqa baGaaG4maaaaaOqaaiaaiwdacqGHxdaTcaaIYaGaaGynaaaaaaa@3FB1@

d.   2 4 ( 7 0 + 3 0 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIYaWaaWbaaSqabeaacaaI0aaaaaGcbaWaaeWaaeaacaaI3aWaaWba aSqabeaaiiaacqWFWaamaaGccqWFRaWkcqWFZaWmdaahaaWcbeqaai ab=bdaWaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaG4maaaaaaaa aa@3ED6@

Solution

(d)

Let us solve each option one by one.

Option a,

2 3 × 3 2 4×18 = 2×2×2×3×3 4×18 = 4×18 4×18 =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaWcaa qaaiacaYiIYaWaiaiJCaaaleqcaYyaiaiJcGaGmI4maaaakiadaYOH xdaTcGaGmI4mamacaYihaaWcbKaGmgacaYOaiaiJikdaaaaakeaaca aI0aGaey41aqRaaGymaiaaiIdaaaaabaGaeyypa0ZaaSaaaeaacaaI YaGaey41aqRaaGOmaiabgEna0kaaikdacqGHxdaTcaaIZaGaey41aq RaaG4maaqaaiaaisdacqGHxdaTcaaIXaGaaGioaaaaaeaacqGH9aqp daWcaaqaaiaaisdacqGHxdaTcaaIXaGaaGioaaqaaiaaisdacqGHxd aTcaaIXaGaaGioaaaaaeaacqGH9aqpcaaIXaaaaaa@6882@

Option b,

[ ( 2 ) 3 × ( 2 ) 4 ]÷ ( 2 ) 7 = [ (2) 3 × (2) 4 ] (2) 7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWada qaamaabmaabaGaeyOeI0IaaGOmaaGaayjkaiaawMcaamaaCaaaleqa baGaaG4maaaakiabgEna0oaabmaabaGaeyOeI0IaaGOmaaGaayjkai aawMcaamaaCaaaleqabaGaaGinaaaaaOGaay5waiaaw2faaiabgEpa 4oaabmaabaGaeyOeI0IaaGOmaaGaayjkaiaawMcaamaaCaaaleqaba GaaG4naaaaaOqaaKaaGjabg2da9OWaaSaaaKaaGfaakmaadmaabaGa aiikaiabgkHiTiaaikdacaGGPaWaaWbaaSqabeaacaaIZaaaaOGaey 41aqRaaiikaiabgkHiTiaaikdacaGGPaWaaWbaaSqabeaacaaI0aaa aaGccaGLBbGaayzxaaaajaaybaGaaiikaiabgkHiTiaaikdacaGGPa GcdaahaaqcbawabeaacaaI3aaaaaaaaaaa@5B4E@

= (2) 3+4 (2) 7 = (2) 7 (2) 7 =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpdaWcaaqaaiaacIcacqGHsislcaaIYaGaaiykamaaCaaaleqabaGa aG4maiabgUcaRiaaisdaaaaakeaacaGGOaGaeyOeI0IaaGOmaiaacM cadaahaaWcbeqaaiaaiEdaaaaaaaGcbaGaeyypa0ZaaSaaaeaacaGG OaGaeyOeI0IaaGOmaiaacMcadaahaaWcbeqaaiaaiEdaaaaakeaaca GGOaGaeyOeI0IaaGOmaiaacMcadaahaaWcbeqaaiaaiEdaaaaaaaGc baGaeyypa0JaaGymaaaaaa@4B51@

Option c,

3 0 × 5 3 5×25  = 1×5×5×5  5×25 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWcaa qaaiaaiodadaahaaWcbeqaaiaaicdaaaGccqGHxdaTcaaI1aWaaWba aSqabeaacaaIZaaaaaGcbaGaaGynaiabgEna0kaaikdacaaI1aaaaa qaaiaacckacqGH9aqpdaWcaaqaaiaaigdacqGHxdaTcaaI1aGaey41 aqRaaGynaiabgEna0kaaiwdaaeaacaGGGcGaaGynaiabgEna0kaaik dacaaI1aaaaaaaaa@50A4@

= 5×25 5×25 =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpdaWcaaqaaiaaiwdacqGHxdaTcaaIYaGaaGynaaqaaiaaiwdacqGH xdaTcaaIYaGaaGynaaaaaeaacqGH9aqpcaaIXaaaaaa@4158@

Option d,

2 4 ( 7 0 + 3 0 ) 3  = 2 4 (1+1) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaWcaa qaaiaaikdadaahaaWcbeqaaiaaisdaaaaakeaacaGGOaGaaG4namaa CaaaleqabaGaaGimaaaakiabgUcaRiaaiodadaahaaWcbeqaaiaaic daaaGccaGGPaWaaWbaaSqabeaacaaIZaaaaaaaaOqaaiaacckacqGH 9aqpdaWcaaqaaiaaikdadaahaaWcbeqaaiaaisdaaaaakeaacaGGOa GaaGymaiabgUcaRiaaigdacaGGPaWaaWbaaSqabeaacaaIZaaaaaaa aaaa@46B4@

= 2 4 2 3   = 2 43 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpdaWcaaqaaiaaikdadaahaaWcbeqaaiaaisdaaaaakeaacaaIYaWa aWbaaSqabeaacaaIZaaaaaaakiaacckaaeaacqGH9aqpcaaIYaWaaW baaSqabeaacaaI0aGaeyOeI0IaaG4maaaaaaaa@3FD1@

= 2 1 =2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcqGHsislcaaIYaWaaWbaaSqabeaacaaIXaaaaaGcbaGaeyypa0Ja aGOmaaaaaa@3B42@

Hence, option (d) is not equal to 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaaaa@3694@ .

Question: 16

( 2 3 ) 3 × ( 5 7 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaikdaaeaacaaIZaaaaaGaayjkaiaawMcaamaaCaaaleqa baGaaG4maaaakiabgEna0oaabmaabaWaaSaaaeaacaaI1aaabaGaaG 4naaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiodaaaaaaa@3FF9@  is equal to

a.    ( 2 3 × 5 7 ) 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaikdaaeaacaaIZaaaaiabgEna0oaalaaabaGaaGynaaqa aiaaiEdaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI5aaaaaaa@3D82@

b.   ( 2 3 × 5 7 ) 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaikdaaeaacaaIZaaaaiabgEna0oaalaaabaGaaGynaaqa aiaaiEdaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI2aaaaaaa@3D7F@

c.    ( 2 3 × 5 7 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaikdaaeaacaaIZaaaaiabgEna0oaalaaabaGaaGynaaqa aiaaiEdaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaaaaaaa@3D7C@

d.   ( 2 3 × 5 7 ) 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaikdaaeaacaaIZaaaaiabgEna0oaalaaabaGaaGynaaqa aiaaiEdaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIWaaaaaaa@3D79@

Solution

(c)

We have studied that, if a, b and m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacY cacaqGGaGaamOyaiaabccacaqGHbGaaeOBaiaabsgacaqGGaGaamyB aaaa@3DEC@  are rational numbers, then a m × b m = (ab) m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaamyBaaaakiabgEna0kaadkgadaahaaWcbeqaaiaad2ga aaGccqGH9aqpcaGGOaGaamyyaiaadkgacaGGPaWaaWbaaSqabeaaca WGTbaaaaaa@4159@

Here,

a= 2 3 , b= 5 7 , m=3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGHb Gaeyypa0ZaaSaaaeaacaaIYaaabaGaaG4maaaacaGGSaaabaGaamOy aiabg2da9maalaaabaGaaGynaaqaaiaaiEdaaaGaaiilaaqaaiaad2 gacqGH9aqpcaaIZaaaaaa@40E7@

Thus,    ( 2 3 ) 3  × ( 5 7 ) 3   = ( 2 3 × 5 7 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaqada qaamaalaaabaGaaGOmaaqaaiaaiodaaaaacaGLOaGaayzkaaWaaWba aSqabeaacaaIZaaaaOGaaiiOaiabgEna0oaabmaabaWaaSaaaeaaca aI1aaabaGaaG4naaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaioda aaGccaGGGcaabaGaeyypa0ZaaeWaaeaadaWcaaqaaiaaikdaaeaaca aIZaaaaiabgEna0oaalaaabaGaaGynaaqaaiaaiEdaaaaacaGLOaGa ayzkaaWaaWbaaSqabeaacaaIZaaaaaaaaa@4AF9@

Question: 17

In standard form, the number 829030000 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaaik dacaaI5aGaaGimaiaaiodacaaIWaGaaGimaiaaicdacaaIWaaaaa@3C79@  is written as K× 10 8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4saiaayk W7cqGHxdaTcaaIXaGaaGimamaaCaaaleqabaGaaGioaaaaaaa@3CAD@  where K is equal to

a.    82903 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaaik dacaaI5aGaaGimaiaaiodaaaa@3991@

b.   829.03 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaaik dacaaI5aGaaiOlaiaaicdacaaIZaaaaa@3A43@

c.    82.903 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaaik dacaGGUaGaaGyoaiaaicdacaaIZaaaaa@3A43@

d.   8.2903 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaac6 cacaaIYaGaaGyoaiaaicdacaaIZaaaaa@3A43@

Solution

(d)

We know that, a number in a standard form is written as K ×  10 8 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey41aqRaae iiaiaaigdacaaIWaWaaWbaaSqabeaacaaI4aaaaOGaaiilaaaa@3BB0@  where K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4saaaa@36A7@  is a terminating decimal such that 1K<10. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgs MiJkaabUeacqGH8aapcaaIXaGaaGimaiaac6caaaa@3C41@  

So, there is only one option, where K=8.2903 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4saiabg2 da9iaaiIdacaGGUaGaaGOmaiaaiMdacaaIWaGaaG4maaaa@3C17@  less then 10. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic dacaGGUaaaaa@3800@

Question: 18

Which of the following has the largest value?

a.    0.0001 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiaac6 cacaaIWaGaaGimaiaaicdacaaIXaaaaa@3A2E@

b.   1 10000 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGymaiaaicdacaaIWaGaaGimaiaaicdaaaaaaa@3A47@

c.    1 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGymaiaaicdadaahaaWcbeqaaiaaiAdaaaaaaaaa@3906@

d.   1 10 6 ÷0.1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGymaiaaicdadaahaaWcbeqaaiaaiAdaaaaaaOGaey49 aGRaaGimaiaac6cacaaIXaaaaa@3D72@

Solution

(a, b)

Let us solve each option one by one.

Option a,   

0.0001 = 1 10000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaaIWa GaaiOlaiaaicdacaaIWaGaaGimaiaaigdaaeaacqGH9aqpdaWcaaqa aiaaigdaaeaacaaIXaGaaGimaiaaicdacaaIWaGaaGimaaaaaaaa@3FA8@

Option b,

1 10000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaeymaiaaicdacaaIWaGaaGimaiaaicdaaaaaaa@3A3F@

Option c,

1 10 6 = 1 10×10×10×10×10×10 = 1 1000000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWcaa qaaiaaigdaaeaacaaIXaGaaGimamaaCaaaleqabaGaaGOnaaaaaaaa keaacqGH9aqpdaWcaaqaaiaaigdaaeaacaaIXaGaaGimaiabgEna0k aaigdacaaIWaGaey41aqRaaGymaiaaicdacqGHxdaTcaaIXaGaaGim aiabgEna0kaaigdacaaIWaGaey41aqRaaGymaiaaicdaaaaabaGaey ypa0ZaaSaaaeaacaaIXaaabaGaaGymaiaaicdacaaIWaGaaGimaiaa icdacaaIWaGaaGimaaaaaaaa@5501@

Option d,

1 10 6 ÷0.1 = 1 10 6 × 1 0.1 = 1 10 6 × 10 1 = 10 10 6 = 1 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWcaa qaaiaaigdaaeaacaaIXaGaaGimamaaCaaaleqabaGaaGOnaaaaaaGc cqGH3daUcaaIWaGaaiOlaiaaigdaaeaacqGH9aqpdaWcaaqaaiaaig daaeaacaaIXaGaaGimamaaCaaaleqabaGaiWiGiAdaaaaaaOGaey41 aq7aaSaaaeaacaaIXaaabaGaaGimaiaac6cacaaIXaaaaaqaaiabg2 da9maalaaabaGaaGymaaqaaiaaigdacaaIWaWaaWbaaSqabeaacGaJ aIOnaaaaaaGccqGHxdaTdaWcaaqaaiaaigdacaaIWaaabaGaaGymaa aaaeaacqGH9aqpdaWcaaqaaiaaigdacaaIWaaabaGaaGymaiaaicda daahaaWcbeqaaiaaiAdaaaaaaaGcbaGaeyypa0ZaaSaaaeaacaaIXa aabaGaaGymaiaaicdadaahaaWcbeqaaiacmciI1aaaaaaaaaaa@5BBA@

= 1 10×10×10×10×10 = 1  100000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpdaWcaaqaaiaaigdaaeaacaaIXaGaaGimaiabgEna0kaaigdacaaI WaGaey41aqRaaGymaiaaicdacqGHxdaTcaaIXaGaaGimaiabgEna0k aaigdacaaIWaaaaaqaaiabg2da9maalaaabaGaaGymaaqaaiaaccka caaIXaGaaGimaiaaicdacaaIWaGaaGimaiaaicdaaaaaaaa@4EA7@

The fraction whose denominator is the smallest will be the largest fraction.

Hence, a and b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaabc cacaqGHbGaaeOBaiaabsgacaqGGaGaamOyaaaa@3BA7@  are the largest.

Question: 19

In standard form 72 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4naiaaik daaaa@3756@  crore is written as

a.    72× 10 7 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4naiaaik dacqGHxdaTcaaIXaGaaGimamaaCaaaleqabaGaaG4naaaaaaa@3BD0@

b.   72× 10 8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4naiaaik dacqGHxdaTcaaIXaGaaGimamaaCaaaleqabaGaaGioaaaaaaa@3BD1@

c.    7.2× 10 8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4naiaac6 cacaaIYaGaey41aqRaaGymaiaaicdadaahaaWcbeqaaiaaiIdaaaaa aa@3C83@

d.   7.2× 10 7 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4naiaac6 cacaaIYaGaey41aqRaaGymaiaaicdadaahaaWcbeqaaiaaiEdaaaaa aa@3C82@

Solution

(c)

We have studied that, a number in standard form is written as K × 10 n , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey41aqRaaG PaVlaaigdacaaIWaWaaWbaaSqabeaacaWGUbaaaOGaaiilaaaa@3CC9@  where K is the terminating decimal such that 1K<10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgs MiJkaabUeacaqG8aGaaGymaiaaicdaaaa@3B4A@  and n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36CC@  is any integer.

Thus, 72 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4naiaaik daaaa@3755@  crore =720000000 =7.2× 10 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaaI3aGaaGOmaiaaicdacaaIWaGaaGimaiaaicdacaaIWaGaaGim aiaaicdaaeaacqGH9aqpcaaI3aGaaiOlaiaaikdacqGHxdaTcaaIXa GaaGimamaaCaaaleqabaGaaGioaaaaaaaa@4527@

Question: 20

For non-zero numbers a and b, ( a b ) m ÷ ( a b ) n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaadggaaeaacaWGIbaaaaGaayjkaiaawMcaamaaCaaaleqa baGaamyBaaaakiabgEpa4oaabmaabaWaaSaaaeaacaWGHbaabaGaam OyaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaad6gaaaaaaa@4128@ , where m>n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabg6 da+iaad6gaaaa@38C6@ , is equal to

a.    ( a b ) mn MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaadggaaeaacaWGIbaaaaGaayjkaiaawMcaamaaCaaaleqa baGaamyBaiaad6gaaaaaaa@3B51@

b.   ( a b ) m+n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaadggaaeaacaWGIbaaaaGaayjkaiaawMcaamaaCaaaleqa baGaamyBaiabgUcaRiaad6gaaaaaaa@3C33@

c.    ( a b ) mn MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaadggaaeaacaWGIbaaaaGaayjkaiaawMcaamaaCaaaleqa baGaamyBaiabgkHiTiaad6gaaaaaaa@3C3D@

d.   [ ( a b ) m ] n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada qadaqaamaalaaabaGaamyyaaqaaiaadkgaaaaacaGLOaGaayzkaaWa aWbaaSqabeaacaWGTbaaaaGccaGLBbGaayzxaaWaaWbaaSqabeaaca WGUbaaaaaa@3D7A@

Solution

(c)

We have studied that, a m  ÷ a n  = a mn , ( m>n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGHbWaaWbaaSqabeaacaWGTbaaaOGaaiiOaiabgEpa4kaadgga daahaaWcbeqaaiaad6gaaaGccaGGGcGaeyypa0JaamyyamaaCaaale qabaGaamyBaiabgkHiTiaad6gaaaGccaGGSaGaaeiia8aadaqadaqa a8qacaWGTbGaeyOpa4JaamOBaaWdaiaawIcacaGLPaaaaaa@4986@

Thus, ( a b ) m ÷ ( a b ) n = ( a b ) mn MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaqada qaamaalaaabaGaamyyaaqaaiaadkgaaaaacaGLOaGaayzkaaWaaWba aSqabeaacaWGTbaaaOGaey49aG7aaeWaaeaadaWcaaqaaiaadggaae aacaWGIbaaaaGaayjkaiaawMcaamaaCaaaleqabaGaamOBaaaaaOqa aiabg2da9maabmaabaWaaSaaaeaacaWGHbaabaGaamOyaaaaaiaawI cacaGLPaaadaahaaWcbeqaaiaad2gacqGHsislcaWGUbaaaaaaaa@48A4@  

Question: 21

Which of the following is not true?

a.    3 2 > 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaCa aaleqabaGaaGOmaaaakiabg6da+iaaikdadaahaaWcbeqaaiaaioda aaaaaa@3A37@

b.   4 3 = 2 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinamaaCa aaleqabaGaaG4maaaakiabg2da9iaaikdadaahaaWcbeqaaiaaiAda aaaaaa@3A3A@

c.    3 3 =9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaCa aaleqabaGaaG4maaaakiabg2da9iaaiMdaaaa@3953@

d.   2 5 > 5 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaGynaaaakiabg6da+iaaiwdadaahaaWcbeqaaiaaikda aaaaaa@3A3B@

Solution

(c)

Let us solve each option one by one.

Option a,

3 2  > 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIZaWaaWbaaSqabeaacaaIYaaaaOGaaiiOaiabg6da+iaaikda daahaaWcbeqaaiaaiodaaaaaaa@3B7A@  

3×3>2×2×2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIZaGaey41aqRaaG4maiabg6da+iaaikdacqGHxdaTcaaIYaGa ey41aqRaaGOmaaaa@40F3@  

9>8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaI5aGaeyOpa4JaaGioaaaa@3885@  (true)

 Option b,

4 3  = 2 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaI0aWaaWbaaSqabeaacaaIZaaaaOGaaiiOaiabg2da9iaaikda daahaaWcbeqaaiaaiAdaaaaaaa@3B7D@  

( 2 2 ) 3  = 2 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaabaaa aaaaaapeGaaGOmamaaCaaaleqabaGaaGOmaaaak8aacaGGPaWdbmaa CaaaleqabaGaaG4maaaakiaacckacqGH9aqpcaaIYaWaaWbaaSqabe aacaaI2aaaaaaa@3DE6@  

2 6  = 2 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIYaWaaWbaaSqabeaacaaI2aaaaOGaaiiOaiabg2da9iaaikda daahaaWcbeqaaiaaiAdaaaaaaa@3B7E@   (true)

Option c,

3 3  =9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIZaWaaWbaaSqabeaacaaIZaGaaiiOaaaakiabg2da9iaaiMda aaa@3A96@  

3×3×3=9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIZaGaey41aqRaaG4maiabgEna0kaaiodacqGH9aqpcaaI5aaa aa@3E26@

279 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIYaGaaG4naiabgcMi5kaaiMdaaaa@39FF@  (false)

Option d,

2 5 > 5 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIYaWaaWbaaSqabeaacaaI1aaaaOGaeyOpa4JaaGynamaaCaaa leqabaGaaGOmaaaaaaa@3A5A@  

2×2×2×2×2=5×5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIYaGaey41aqRaaGOmaiabgEna0kaaikdacqGHxdaTcaaIYaGa ey41aqRaaGOmaiabg2da9iaaiwdacqGHxdaTcaaI1aaaaa@469B@  

32>25 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIZaGaaGOmaiabg6da+iaaikdacaaI1aaaaa@39F4@  (true)

Hence, option (c) is not true.

Question: 22

Which power of 8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaaaa@369B@  is equal to 2 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaGOnaaaaaaa@3782@ ?

a.    3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maaaa@3696@

b.   2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaaaa@3695@

c.    1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaaaa@3694@

d.   4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaaaa@3697@

Solution

(b)

Let us suppose that the power of 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaI4aaaaa@36BA@  be x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG4baaaa@36F5@ .

According to the question, we have

8 x = 2 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaI4aWaaWbaaSqabeaacaWG4baaaOGaeyypa0JaaGOmamaaCaaa leqabaGaaGOnaaaaaaa@3A9D@  

( 2 3 ) x = 2 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaabaaa aaaaaapeGaaGOmamaaCaaaleqabaGaaG4maaaak8aacaGGPaWdbmaa CaaaleqabaGaamiEaaaakiabg2da9iaaikdadaahaaWcbeqaaiaaiA daaaaaaa@3D03@  

2 3x = 2 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIYaWaaWbaaSqabeaacaaIZaGaamiEaaaakiabg2da9iaaikda daahaaWcbeqaaiaaiAdaaaaaaa@3B54@  

Since, bases are equal, by equating their exponents, we get

3x=6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIZaGaamiEaiabg2da9iaaiAdaaaa@3978@  

3x 3 = 6 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWcaaqaaiaaiodacaWG4baabaGaaG4maaaacqGH9aqpdaWcaaqa aiaaiAdaaeaacaaIZaaaaaaa@3B12@  

x=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG4bGaeyypa0JaaGOmaaaa@38B7@  

Hence, the power of 8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaaaa@369B@  is 2, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaacY caaaa@3745@  which is equal to 2 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIYaWaaWbaaSqabeaacaaI2aaaaaaa@37A1@ .

 

 

In questions 23 to 39, fill in the blanks to make the statements true.

 

Question: 23

( 2 ) 31 × ( 2 ) 13 = ( 2 ) ? MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq GHsislcaaIYaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaGaaGym aaaakiabgEna0oaabmaabaGaeyOeI0IaaGOmaaGaayjkaiaawMcaam aaCaaaleqabaGaaGymaiaaiodaaaGccqGH9aqpdaqadaqaaiabgkHi TiaaikdaaiaawIcacaGLPaaadaahaaWcbeqaaiaac+daaaaaaa@46D9@

Solution

Here,

  ( 2 ) 31 × ( 2 ) 13 (a) m × (a) n = a m+n = ( 2 ) 31+13 = ( 2 ) 44 ( 2 ) 31 × ( 2 ) 13 = ( 2 ) 44 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaqada qaaiabgkHiTiaaikdaaiaawIcacaGLPaaadaahaaWcbeqaaiaaioda caaIXaaaaOGaey41aq7aaeWaaeaacqGHsislcaaIYaaacaGLOaGaay zkaaWaaWbaaSqabeaacaaIXaGaaG4maaaaaOqaaabaaaaaaaaapeGa eSynIeLaaeikaiaabggacaqGPaWaaWbaaSqabeaacaqGTbaaaOGaey 41aqRaaeikaiaabggacaqGPaWaaWbaaSqabeaacaWGUbaaaOGaeyyp a0JaaeyyamaaCaaaleqabaGaaeyBaiaabUcacaqGUbaaaaGcbaGaey ypa0ZdamaabmaabaGaeyOeI0IaaGOmaaGaayjkaiaawMcaamaaCaaa leqabaGaaG4maiaaigdacqGHRaWkcaaIXaGaaG4maaaaaOqaa8qacq GH9aqppaWaaeWaaeaacqGHsislcaaIYaaacaGLOaGaayzkaaWaaWba aSqabeaacaaI0aGaaGinaaaaaOqaaiabgsJiCnaabmaabaGaeyOeI0 IaaGOmaaGaayjkaiaawMcaamaaCaaaleqabaGaaG4maiaaigdaaaGc cqGHxdaTdaqadaqaaiabgkHiTiaaikdaaiaawIcacaGLPaaadaahaa WcbeqaaiaaigdacaaIZaaaaaGcbaGaeyypa0ZaaeWaaeaacqGHsisl caaIYaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI0aGaaGinaaaaaa aa@71F4@  

Question: 24

( 3 ) 8 ÷ ( 3 ) 5 = ( 3 ) ? MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq GHsislcaaIZaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI4aaaaOGa ey49aG7aaeWaaeaacqGHsislcaaIZaaacaGLOaGaayzkaaWaaWbaaS qabeaacaaI1aaaaOGaeyypa0ZaaeWaaeaacqGHsislcaaIZaaacaGL OaGaayzkaaWaaWbaaSqabeaacaGG=aaaaaaa@4592@

Solution

Here,

  ( 3 ) 8 ÷ ( 3 ) 5 a m ÷ a n = a mn = ( 3 ) 85 = ( 3 ) 3 ( 3 ) 8 ÷ ( 3 ) 5 = ( 3 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaqada qaaiabgkHiTiaaiodaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiIda aaGccqGH3daUdaqadaqaaiabgkHiTiaaiodaaiaawIcacaGLPaaada ahaaWcbeqaaiaaiwdaaaaakeaaqaaaaaaaaaWdbiablwJirjaadgga daahaaWcbeqaaiaad2gaaaGccqGH3daUcaWGHbWaaWbaaSqabeaaca WGUbaaaOGaeyypa0JaamyyamaaCaaaleqabaGaamyBaiabgkHiTiaa d6gaaaaakeaacqGH9aqppaWaaeWaaeaacqGHsislcaaIZaaacaGLOa GaayzkaaWaaWbaaSqabeaacaaI4aGaeyOeI0IaaGynaaaaaOWdbeaa cqGH9aqppaWaaeWaaeaacqGHsislcaaIZaaacaGLOaGaayzkaaWaaW baaSqabeaacaaIZaaaaaGcbaGaeyinIW1aaeWaaeaacqGHsislcaaI ZaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI4aaaaOGaey49aG7aae WaaeaacqGHsislcaaIZaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI 1aaaaaGcbaGaeyypa0ZaaeWaaeaacqGHsislcaaIZaaacaGLOaGaay zkaaWaaWbaaSqabeaacaaIZaaaaaaaaa@6A44@  

 

Question: 25

( 11 15 ) 4 × ( ? ) 5 = ( 11 15 ) 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaigdacaaIXaaabaGaaGymaiaaiwdaaaaacaGLOaGaayzk aaWaaWbaaSqabeaacaaI0aaaaOGaey41aq7aaeWaaeaacaGG=aaaca GLOaGaayzkaaWaaWbaaSqabeaacaaI1aaaaOGaeyypa0ZaaeWaaeaa daWcaaqaaiaaigdacaaIXaaabaGaaGymaiaaiwdaaaaacaGLOaGaay zkaaWaaWbaaSqabeaacaaI5aaaaaaa@472F@

Solution

Let us suppose that

( 11 15 ) 4 × (x) 5 = ( 11 15 ) 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaqadaqaamaalaaabaGaaGymaiaaigdaaeaacaaIXaGaaGynaaaa aiaawIcacaGLPaaadaahaaWcbeqaaiaaisdaaaGccqGHxdaTpaGaai ika8qacaWG4bWdaiaacMcapeWaaWbaaSqabeaacaaI1aaaaOGaeyyp a0ZaaeWaaeaadaWcaaqaaiaaigdacaaIXaaabaGaaGymaiaaiwdaaa aacaGLOaGaayzkaaWaaWbaaSqabeaacaaI5aaaaaaa@4796@  

(x) 5  = ( 11 15 ) 9 ( 11 15 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taai ikaabaaaaaaaaapeGaamiEa8aacaGGPaWdbmaaCaaaleqabaGaaGyn aaaakiaacckacqGH9aqpdaWcaaqaamaabmaabaWaaSaaaeaacaaIXa GaaGymaaqaaiaaigdacaaI1aaaaaGaayjkaiaawMcaamaaCaaaleqa baGaaGyoaaaaaOqaamaabmaabaWaaSaaaeaacaaIXaGaaGymaaqaai aaigdacaaI1aaaaaGaayjkaiaawMcaamaaCaaaleqabaGaaGinaaaa aaaaaa@48F1@

(x) 5  = ( 11 15 ) 9 ÷ ( 11 15 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taai ikaabaaaaaaaaapeGaamiEa8aacaGGPaWdbmaaCaaaleqabaGaaGyn aaaakiaacckacqGH9aqpdaqadaqaamaalaaabaGaaGymaiaaigdaae aacaaIXaGaaGynaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiMda aaGccqGH3daUdaqadaqaamaalaaabaGaaGymaiaaigdaaeaacaaIXa GaaGynaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaisdaaaaaaa@4B1C@

(x) 5  = ( 11 15 ) 9 4                     a m ÷ a n = a mn MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taai ikaabaaaaaaaaapeGaamiEa8aacaGGPaWdbmaaCaaaleqabaGaaGyn aaaakiaacckacqGH9aqpdaqadaqaamaalaaabaGaaGymaiaaigdaae aacaaIXaGaaGynaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiMda aaGcdaahaaWcbeqaaiabgkHiTiaaisdaaaGccaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaeSynIeLaamyyamaaCaaaleqabaGaamyBaaaakiabgEpa4kaadgga daahaaWcbeqaaiaad6gaaaGccqGH9aqpcaWGHbWaaWbaaSqabeaaca WGTbGaeyOeI0IaamOBaaaaaaa@5DE6@

(x) 5  = ( 11 15 ) 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taai ikaabaaaaaaaaapeGaamiEa8aacaGGPaWdbmaaCaaaleqabaGaaGyn aaaakiaacckacqGH9aqpdaqadaqaamaalaaabaGaaGymaiaaigdaae aacaaIXaGaaGynaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiwda aaaaaa@435F@   

Since, in the above equation, the powers are same. Thus, x= 11 15 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG4bGaeyypa0ZaaSaaaeaacaaIXaGaaGymaaqaaiaaigdacaaI 1aaaaaaa@3AFB@  

Hence, ( 11 15 ) 4 × ( 11 15 ) 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaqadaqaamaalaaabaGaaGymaiaaigdaaeaacaaIXaGaaGynaaaa aiaawIcacaGLPaaadaahaaWcbeqaaiaaisdaaaGccqGHxdaTdaqada qaamaalaaabaGaaGymaiaaigdaaeaacaaIXaGaaGynaaaaaiaawIca caGLPaaadaahaaWcbeqaaiaaiwdaaaaaaa@4302@ = ( 11 15 ) 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGH9aqpdaqadaqaamaalaaabaGaaGymaiaaigdaaeaacaaIXaGa aGynaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiMdaaaaaaa@3C77@

Question: 26

( 1 4 ) 3 × ( 1 4 ) ? = ( 1 4 ) 11 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiabgkHiTiaaigdaaeaacaaI0aaaaaGaayjkaiaawMcaamaa CaaaleqabaGaaG4maaaakiabgEna0oaabmaabaWaaSaaaeaacqGHsi slcaaIXaaabaGaaGinaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaa c+daaaGccqGH9aqpdaqadaqaamaalaaabaGaeyOeI0IaaGymaaqaai aaisdaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIXaGaaGymaaaa aaa@4884@

Solution

Let us suppose that

( 1 4 ) 3 × ( 1 4 ) x = ( 1 4 ) 11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaqadaqaaiabgkHiTmaalaaabaGaaGymaaqaaiaaisdaaaaacaGL OaGaayzkaaWaaWbaaSqabeaacaaIZaaaaOGaey41aq7aaeWaaeaacq GHsisldaWcaaqaaiaaigdaaeaacaaI0aaaaaGaayjkaiaawMcaamaa CaaaleqabaGaamiEaaaakiabg2da9maabmaabaGaeyOeI0YaaSaaae aacaaIXaaabaGaaGinaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaa igdacaaIXaaaaaaa@48DD@

( 1 4 ) x = ( 1 4 ) 11 ( 1 4 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaqadaqaaiabgkHiTmaalaaabaGaaGymaaqaaiaaisdaaaaacaGL OaGaayzkaaWaaWbaaSqabeaacaWG4baaaOGaeyypa0ZaaSaaaeaada qadaqaaiabgkHiTmaalaaabaGaaGymaaqaaiaaisdaaaaacaGLOaGa ayzkaaWaaWbaaSqabeaacaaIXaGaaGymaaaaaOqaamaabmaabaGaey OeI0YaaSaaaeaacaaIXaaabaGaaGinaaaaaiaawIcacaGLPaaadaah aaWcbeqaaiaaiodaaaaaaaaa@46D6@

( 1 4 ) x = ( 1 4 ) 113                      a m ÷ a n = a mn MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaqadaqaaiabgkHiTmaalaaabaGaaGymaaqaaiaaisdaaaaacaGL OaGaayzkaaWaaWbaaSqabeaacaWG4baaaOGaeyypa0ZaaeWaaeaacq GHsisldaWcaaqaaiaaigdaaeaacaaI0aaaaaGaayjkaiaawMcaamaa CaaaleqabaGaaGymaiaaigdacqGHsislcaaIZaaaaOGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacqWI1isucaWGHbWaaWbaaSqabeaacaWGTbaaaOGa ey49aGRaamyyamaaCaaaleqabaGaamOBaaaakiabg2da9iaadggada ahaaWcbeqaaiaad2gacqGHsislcaWGUbaaaaaa@5CC1@

( 1 4 ) x = ( 1 4 ) 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaqadaqaaiabgkHiTmaalaaabaGaaGymaaqaaiaaisdaaaaacaGL OaGaayzkaaWaaWbaaSqabeaacaWG4baaaOGaeyypa0ZaaeWaaeaacq GHsisldaWcaaqaaiaaigdaaeaacaaI0aaaaaGaayjkaiaawMcaamaa CaaaleqabaGaaGioaaaaaaa@411F@

Since in the above equation the bases are equal. So, by equating the powers, we get, x=8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 da9iaaiIdaaaa@389E@  

Hence, ( 1 4 ) 3 × ( 1 4 ) 8 = ( 1 4 ) 11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaqadaqaaiabgkHiTmaalaaabaGaaGymaaqaaiaaisdaaaaacaGL OaGaayzkaaWaaWbaaSqabeaacaaIZaaaaOGaey41aq7aaeWaaeaacq GHsisldaWcaaqaaiaaigdaaeaacaaI0aaaaaGaayjkaiaawMcaamaa CaaaleqabaGaaGioaaaakiabg2da9maabmaabaGaeyOeI0YaaSaaae aacaaIXaaabaGaaGinaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaa igdacaaIXaaaaaaa@48A2@

Question: 27

[ ( 7 11 ) 3 ] 4 = ( 7 11 ) ? MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada qadaqaamaalaaabaGaaG4naaqaaiaaigdacaaIXaaaaaGaayjkaiaa wMcaamaaCaaaleqabaGaaG4maaaaaOGaay5waiaaw2faamaaCaaale qabaGaaGinaaaakiabg2da9maabmaabaWaaSaaaeaacaaI3aaabaGa aGymaiaaigdaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaGG=aaaaa aa@4349@

Solution

Here, [ ( 7 11 ) 3 ] 4 = ( 7 11 ) 3×4 = ( 7 11 ) 12     ( a m ) n = a mn MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWadaqaamaabmaabaWaaSaaaeaacaaI3aaabaGaaGymaiaaigda aaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaaaaaGccaGLBbGaay zxaaWaaWbaaSqabeaacaaI0aaaaOGaeyypa0ZaaeWaaeaadaWcaaqa aiaaiEdaaeaacaaIXaGaaGymaaaaaiaawIcacaGLPaaadaahaaWcbe qaaiaaiodacqGHxdaTcaaI0aaaaOGaeyypa0ZaaeWaaeaadaWcaaqa aiaaiEdaaeaacaaIXaGaaGymaaaaaiaawIcacaGLPaaadaahaaWcbe qaaiaaigdacaaIYaaaaOGaaeiiaiaabccacaqGGaGaeSynIeLaaiik aiaadggadaahaaWcbeqaaiaad2gaaaGccaGGPaWaaWbaaSqabeaaca WGUbaaaOGaeyypa0JaamyyamaaCaaaleqabaGaamyBaiaad6gaaaaa aa@5878@

        [ ( 7 11 ) 3 ] 4 = ( 7 11 ) 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGH0icxcaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaWaamWaaeaadaqadaqaamaalaaabaGaaG4naaqaaiaaigdacaaIXa aaaaGaayjkaiaawMcaamaaCaaaleqabaGaaG4maaaaaOGaay5waiaa w2faamaaCaaaleqabaGaaGinaaaakiabg2da9maabmaabaWaaSaaae aacaaI3aaabaGaaGymaiaaigdaaaaacaGLOaGaayzkaaWaaWbaaSqa beaacaaIXaGaaGOmaaaaaaa@49D0@

Question: 28

( 6 13 ) 10 ÷ [ ( 6 13 ) 5 ] 2 = ( 6 13 ) ? MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaiAdaaeaacaaIXaGaaG4maaaaaiaawIcacaGLPaaadaah aaWcbeqaaiaaigdacaaIWaaaaOGaey49aG7aamWaaeaadaqadaqaam aalaaabaGaaGOnaaqaaiaaigdacaaIZaaaaaGaayjkaiaawMcaamaa CaaaleqabaGaaGynaaaaaOGaay5waiaaw2faamaaCaaaleqabaGaaG Omaaaakiabg2da9maabmaabaWaaSaaaeaacaaI2aaabaGaaGymaiaa iodaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaGG=aaaaaaa@4B04@

Solution

( 6 13 ) 10 ÷ [ ( 6 13 ) 5 ] 2                               ( a m ) n = a mn MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaiAdaaeaacaaIXaGaaG4maaaaaiaawIcacaGLPaaadaah aaWcbeqaaiaaigdacaaIWaaaaOGaey49aG7aamWaaeaadaqadaqaam aalaaabaGaaGOnaaqaaiaaigdacaaIZaaaaaGaayjkaiaawMcaamaa CaaaleqabaGaaGynaaaaaOGaay5waiaaw2faamaaCaaaleqabaGaaG OmaaaakiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaaeaaaaaaaaa8qacq WI1isucaGGOaGaamyyamaaCaaaleqabaGaamyBaaaakiaacMcadaah aaWcbeqaaiaad6gaaaGccqGH9aqpcaWGHbWaaWbaaSqabeaacaWGTb GaamOBaaaaaaa@619D@

= ( 6 13 ) 10 ÷ ( 6 13 ) 5×2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaadaWcaaqaaiaaiAdaaeaacaaIXaGaaG4maaaaaiaawIcacaGL PaaadaahaaWcbeqaaiaaigdacaaIWaaaaOGaey49aG7aaeWaaeaada WcaaqaaiaaiAdaaeaacaaIXaGaaG4maaaaaiaawIcacaGLPaaadaah aaWcbeqaaiaaiwdacqGHxdaTcaaIYaaaaaaa@4627@

= ( 6 13 ) 10 ÷ ( 6 13 ) 10                              a m ÷ a n = a mn MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaadaWcaaqaaiaaiAdaaeaacaaIXaGaaG4maaaaaiaawIcacaGL PaaadaahaaWcbeqaaiaaigdacaaIWaaaaOGaey49aG7aiWiGbmaabG aJaoacmc4caaqaiWiGcGaJaIOnaaqaiWiGcGaJaIymaiacmciIZaaa aaGaiWiGwIcacGaJaAzkaaWaiWiGCaaaleqcmcyaiWiGcGaJaIymai acmciIWaaaaOGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaabaaaaaaaaapeGaeS ynIeLaamyyamaaCaaaleqabaGaamyBaaaakiabgEpa4kaadggadaah aaWcbeqaaiaad6gaaaGccqGH9aqpcaWGHbWaaWbaaSqabeaacaWGTb GaeyOeI0IaamOBaaaaaaa@72B2@

= ( 6 13 ) 1010   MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaadaWcaaqaaiaaiAdaaeaacaaIXaGaaG4maaaaaiaawIcacaGL PaaadaahaaWcbeqaaiaaigdacaaIWaGaeyOeI0IaaGymaiaaicdaaa GccaqGGaaaaa@3F61@

= ( 6 13 ) 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaadaWcaaqaaiaaiAdaaeaacaaIXaGaaG4maaaaaiaawIcacaGL PaaadaahaaWcbeqaaiaaicdaaaaaaa@3B97@

       ( 6 13 ) 10 ÷ [ ( 6 13 ) 5 ] 2 = ( 6 13 ) 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH0i cxcaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccadaqadaqaamaa laaabaGaaGOnaaqaaiaaigdacaaIZaaaaaGaayjkaiaawMcaamaaCa aaleqabaGaaGymaiaaicdaaaGccqGH3daUdaWadaqaamaabmaabaWa aSaaaeaacaaI2aaabaGaaGymaiaaiodaaaaacaGLOaGaayzkaaWaaW baaSqabeaacaaI1aaaaaGccaGLBbGaayzxaaWaaWbaaSqabeaacaaI YaaaaaGcbaGaeyypa0ZaaeWaaeaadaWcaaqaaiaaiAdaaeaacaaIXa GaaG4maaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaicdaaaaaaaa@5011@

Question: 29

[ ( 1 4 ) 16 ] 2 = ( 1 4 ) ? MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada qadaqaamaalaaabaGaeyOeI0IaaGymaaqaaiaaisdaaaaacaGLOaGa ayzkaaWaaWbaaSqabeaacaaIXaGaaGOnaaaaaOGaay5waiaaw2faam aaCaaaleqabaGaaGOmaaaakiabg2da9maabmaabaWaaSaaaeaacqGH sislcaaIXaaabaGaaGinaaaaaiaawIcacaGLPaaadaahaaWcbeqaai aac+daaaaaaa@4464@

Solution

Here,

[ ( 1 4 ) 16 ] 2                                ( a m ) n = a mn MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWadaqaamaabmaabaWaaSaaaeaacqGHsislcaaIXaaabaGaaGin aaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaigdacaaI2aaaaaGcca GLBbGaayzxaaWaaWbaaSqabeaacaaIYaaaaOGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaeSynIeLaaiikaiaadggadaahaaWcbeqa aiaad2gaaaGccaGGPaWaaWbaaSqabeaacaWGUbaaaOGaeyypa0Jaam yyamaaCaaaleqabaGaamyBaiaad6gaaaaaaa@5B71@

= ( 1 4 ) 16×2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGH9aqpdaqadaqaaiabgkHiTmaalaaabaGaaGymaaqaaiaaisda aaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIXaGaaGOnaiabgEna0k aaikdaaaaaaa@3F78@

= ( 1 4 ) 32 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGH9aqpdaqadaqaaiabgkHiTmaalaaabaGaaGymaaqaaiaaisda aaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaGaaGOmaaaaaaa@3CA3@

[ ( 1 4 ) 16 ] 2 = ( 1 4 ) 32 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGH0icxdaWadaqaamaabmaabaWaaSaaaeaacqGHsislcaaIXaaa baGaaGinaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaigdacaaI2a aaaaGccaGLBbGaayzxaaWaaWbaaSqabeaacaaIYaaaaOGaeyypa0Za aeWaaeaacqGHsisldaWcaaqaaiaaigdaaeaacaaI0aaaaaGaayjkai aawMcaamaaCaaaleqabaGaaG4maiaaikdaaaaaaa@4677@

 

Question: 30

( 13 14 ) 5 ÷ ( ? ) 2 = ( 13 14 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaigdacaaIZaaabaGaaGymaiaaisdaaaaacaGLOaGaayzk aaWaaWbaaSqabeaacaaI1aaaaOGaey49aG7aaeWaaeaacaGG=aaaca GLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaeyypa0ZaaeWaaeaa daWcaaqaaiaaigdacaaIZaaabaGaaGymaiaaisdaaaaacaGLOaGaay zkaaWaaWbaaSqabeaacaaIZaaaaaaa@474D@

Solution

Let us suppose that

  ( 13 14 ) 5 ÷ (x) 2 = ( 13 14 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaqadaqaamaalaaabaGaaGymaiaaiodaaeaacaaIXaGaaGinaaaa aiaawIcacaGLPaaadaahaaWcbeqaaiaaiwdaaaGccqGH3daUcaGGOa GaamiEaiaacMcadaahaaWcbeqaaiaaikdaaaGccqGH9aqpdaqadaqa amaalaaabaGaaGymaiaaiodaaeaacaaIXaGaaGinaaaaaiaawIcaca GLPaaadaahaaWcbeqaaiaaiodaaaaaaa@4776@

( 13 14 ) 5 × 1 (x) 2 = ( 13 14 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaqadaqaamaalaaabaGaaGymaiaaiodaaeaacaaIXaGaaGinaaaa aiaawIcacaGLPaaadaahaaWcbeqaaiaaiwdaaaGccqGHxdaTdaWcaa qaaiaaigdaaeaacaGGOaGaamiEaiaacMcadaahaaWcbeqaaiaaikda aaaaaOGaeyypa0ZaaeWaaeaadaWcaaqaaiaaigdacaaIZaaabaGaaG ymaiaaisdaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaaaaaaa @481D@

(x) 2 = ( 13 14 ) 5 ( 13 14 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaGGOaGaamiEaiaacMcadaahaaWcbeqaaiaaikdaaaGccqGH9aqp daWcaaqaamaabmaabaWaaSaaaeaacaaIXaGaaG4maaqaaiaaigdaca aI0aaaaaGaayjkaiaawMcaamaaCaaaleqabaGaaGynaaaaaOqaamaa bmaabaWaaSaaaeaacaaIXaGaaG4maaqaaiaaigdacaaI0aaaaaGaay jkaiaawMcaamaaCaaaleqabaGaaG4maaaaaaaaaa@454B@

(x) 2 = ( 13 14 ) 53 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaGGOaGaamiEaiaacMcadaahaaWcbeqaaiaaikdaaaGccqGH9aqp daqadaqaamaalaaabaGaaGymaiaaiodaaeaacaaIXaGaaGinaaaaai aawIcacaGLPaaadaahaaWcbeqaaiaaiwdacqGHsislcaaIZaaaaaaa @4167@

(x) 2 = ( 13 14 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaGGOaGaamiEaiaacMcadaahaaWcbeqaaiaaikdaaaGccqGH9aqp daqadaqaamaalaaabaGaaGymaiaaiodaaeaacaaIXaGaaGinaaaaai aawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaaaa@3FBA@

Since in the above equation, the powers are same.

Thus, x= 13 14 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG4bGaeyypa0ZaaSaaaeaacaaIXaGaaG4maaqaaiaaigdacaaI 0aaaaaaa@3AFC@

Hence, ( 13 14 ) 5 ÷ ( 13 14 ) 2 = ( 13 14 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaqadaqaamaalaaabaGaaGymaiaaiodaaeaacaaIXaGaaGinaaaa aiaawIcacaGLPaaadaahaaWcbeqaaiaaiwdaaaGccqGH3daUdaqada qaamaalaaabaGaaGymaiaaiodaaeaacaaIXaGaaGinaaaaaiaawIca caGLPaaadaahaaWcbeqaaiaaikdaaaGccqGH9aqpdaqadaqaamaala aabaGaaGymaiaaiodaaeaacaaIXaGaaGinaaaaaiaawIcacaGLPaaa daahaaWcbeqaaiaaiodaaaaaaa@49AA@

Question: 31

a 6 × a 5 × a 0 = a ? MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaGOnaaaakiabgEna0kaadggadaahaaWcbeqaaiaaiwda aaGccqGHxdaTcaWGHbWaaWbaaSqabeaacaaIWaaaaOGaeyypa0Jaam yyamaaCaaaleqabaGaai4paaaaaaa@4272@

Solution

The expression can be written as a 6 × a 5 × a 0 = a 6+5+0 = a 11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGHbWaaWbaaSqabeaacaaI2aaaaOGaey41aqRaamyyamaaCaaa leqabaGaaGynaaaakiabgEna0kaadggadaahaaWcbeqaaiaaicdaaa GccqGH9aqpcaWGHbWaaWbaaSqabeaacaaI2aGaey4kaSIaaGynaiab gUcaRiaaicdaaaGccqGH9aqpcaWGHbWaaWbaaSqabeaacaaIXaGaaG ymaaaaaaa@4965@  

           a 6 × a 5 × a 0 = a 11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGH0icxcaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaadggadaah aaWcbeqaaiaaiAdaaaGccqGHxdaTcaWGHbWaaWbaaSqabeaacaaI1a aaaOGaey41aqRaamyyamaaCaaaleqabaGaaGimaaaakiabg2da9iaa dggadaahaaWcbeqaaiaaigdacaaIXaaaaaaa@4A37@  

Question: 32

1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaaaa@3694@  lakh = 10 ? MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ymaiaaicdadaahaaWcbeqaaiaac+daaaaaaa@3943@

Solution

We know that,

1 lakh =100000 = 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaaqaaaaa aaaaWdbiaaigdacaqGGaGaaeiBaiaabggacaqGRbGaaeiAaaqaaiab g2da9iaaigdacaaIWaGaaGimaiaaicdacaaIWaGaaGimaaqaaiabg2 da9iaaigdacaaIWaWaaWbaaSqabeaacaaI1aaaaaaaaa@43D3@

 1 lakh= 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGH0icxcaqGGaGaaGymaiaabccacaqGSbGaaeyyaiaabUgacaqG ObGaeyypa0JaaGymaiaaicdadaahaaWcbeqaaiaaiwdaaaaaaa@404A@  

Question: 33

1 million = 10 ? MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ymaiaaicdadaahaaWcbeqaaiaac+daaaaaaa@3944@

Solution

We know that,

1 million =1000000 = 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaaqaaaaa aaaaWdbiaaigdacaqGGaGaaeyBaiaabMgacaqGSbGaaeiBaiaabMga caqGVbGaaeOBaaqaaiabg2da9iaaigdacaaIWaGaaGimaiaaicdaca aIWaGaaGimaiaaicdaaeaacqGH9aqpcaaIXaGaaGimamaaCaaaleqa baGaaGOnaaaaaaaa@476B@  

   1 million= 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGH0icxcaqGGaGaaeiiaiaabccacaaIXaGaaeiiaiaab2gacaqG PbGaaeiBaiaabYgacaqGPbGaae4Baiaab6gacqGH9aqpcaaIXaGaaG imamaaCaaaleqabaGaaGOnaaaaaaa@446E@  

Question: 34

729= 3 ? MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4naiaaik dacaaI5aGaeyypa0JaaG4mamaaCaaaleqabaGaai4paaaaaaa@3ACC@

Solution

Here, we have to find out the factors of the given expression.

So, 729=3×3×3×3×3×3= 3 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaI3aGaaGOmaiaaiMdacqGH9aqpcaaIZaGaey41aqRaaG4maiab gEna0kaaiodacqGHxdaTcaaIZaGaey41aqRaaG4maiabgEna0kaaio dacqGH9aqpcaaIZaWaaWbaaSqabeaacaaI2aaaaaaa@4ACF@

3

729

3

243

3

81

3

27

3

9

3

3

 

1

 729= 3 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIWLaae iiaiaaiEdacaaIYaGaaGyoaiabg2da9iaaiodadaahaaWcbeqaaiaa iAdaaaaaaa@3CA9@

Question: 35

432= 2 4 × 3 ? MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiaaio dacaaIYaGaeyypa0JaaGOmamaaCaaaleqabaGaaGinaaaakiabgEna 0kaaiodadaahaaWcbeqaaiaac+daaaaaaa@3E8A@

Solution

Here, we are suppose to find out the factors of given expression.

So, 432=2×2×2×2×3×3×3= 2 4 × 3 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaI0aGaaG4maiaaikdacqGH9aqpcaaIYaGaey41aqRaaGOmaiab gEna0kaaikdacqGHxdaTcaaIYaGaey41aqRaaG4maiabgEna0kaaio dacqGHxdaTcaaIZaGaeyypa0JaaGOmamaaCaaaleqabaGaaGinaaaa kiabgEna0kaaiodadaahaaWcbeqaaiaaiodaaaaaaa@515B@  

   432= 2 4 × 3 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGH0icxcaqGGaGaaeiiaiaabccacaaI0aGaaG4maiaaikdacqGH 9aqpcaaIYaWaaWbaaSqabeaacaaI0aaaaOGaey41aqRaaG4mamaaCa aaleqabaGaaG4maaaaaaa@41CB@  

Question: 36

53700000=?×1 0 7 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaiaaio dacaaI3aGaaGimaiaaicdacaaIWaGaaGimaiaaicdacqGH9aqpcaaM c8Uaai4paiabgEna0IGaaiab=fdaXiab=bdaWmaaCaaaleqabaGae8 3naCdaaaaa@4419@

Solution

The given number =    53700000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaI1aGaaG4maiaaiEdacaaIWaGaaGimaiaaicdacaaIWaGaaGim aaaa@3BD7@

In standard form, it can be written as 53700000=537× 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaI1aGaaG4maiaaiEdacaaIWaGaaGimaiaaicdacaaIWaGaaGim aiabg2da9iaaiwdacaaIZaGaaG4naiabgEna0kaaigdacaaIWaWaaW baaSqabeaacaaI1aaaaaaa@4392@

Also, 537=5.37× 10 2 × 10 5 =5.37× 10 7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaI1aGaaG4maiaaiEdacqGH9aqpcaaI1aGaaiOlaiaaiodacaaI 3aGaey41aqRaaGymaiaaicdadaahaaWcbeqaaiaaikdaaaGccqGHxd aTcaaIXaGaaGimamaaCaaaleqabaGaaGynaaaakiabg2da9iaaiwda caGGUaGaaG4maiaaiEdacqGHxdaTcaaIXaGaaGimamaaCaaaleqaba GaaG4naaaaaaa@4D9A@  

53700000=5.37× 10 7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaI1aGaaG4maiaaiEdacaaIWaGaaGimaiaaicdacaaIWaGaaGim aiabg2da9iaaiwdacaGGUaGaaG4maiaaiEdacqGHxdaTcaaIXaGaaG imamaaCaaaleqabaGaaG4naaaaaaa@4446@  

Question: 37

88880000000=?× 10 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaaiI dacaaI4aGaaGioaiaaicdacaaIWaGaaGimaiaaicdacaaIWaGaaGim aiaaicdacqGH9aqpcaaMc8Uaai4paiaaykW7cqGHxdaTcaaMc8UaaG ymaiaaicdadaahaaWcbeqaaiaaigdacaaIWaaaaaaa@498F@

Solution

The given number =   88880000000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaI4aGaaGioaiaaiIdacaaI4aGaaGimaiaaicdacaaIWaGaaGim aiaaicdacaaIWaGaaGimaaaa@3E16@  

In standard form, it can be written as 88880000000=8888× 10 7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaI4aGaaGioaiaaiIdacaaI4aGaaGimaiaaicdacaaIWaGaaGim aiaaicdacaaIWaGaaGimaiabg2da9iaaiIdacaaI4aGaaGioaiaaiI dacqGHxdaTcaaIXaGaaGimamaaCaaaleqabaGaaG4naaaaaaa@469E@  

Also, 8888=8.888× 10 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaI4aGaaGioaiaaiIdacaaI4aGaeyypa0JaaGioaiaac6cacaaI 4aGaaGioaiaaiIdacqGHxdaTcaaIXaGaaGimamaaCaaaleqabaGaaG 4maaaaaaa@4236@  

So, 8.888× 10 3 × 10 7 =8.888× 10 10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaI4aGaaiOlaiaaiIdacaaI4aGaaGioaiabgEna0kaaigdacaaI WaWaaWbaaSqabeaacaaIZaaaaOGaey41aqRaaGymaiaaicdadaahaa WcbeqaaiaaiEdaaaGccqGH9aqpcaaI4aGaaiOlaiaaiIdacaaI4aGa aGioaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaIXaGaaGimaa aaaaa@4CA4@  

88880000000=8.888× 10 10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaI4aGaaGioaiaaiIdacaaI4aGaaGimaiaaicdacaaIWaGaaGim aiaaicdacaaIWaGaaGimaiabg2da9iaaiIdacaGGUaGaaGioaiaaiI dacaaI4aGaey41aqRaaGymaiaaicdadaahaaWcbeqaaiaaigdacaaI Waaaaaaa@4804@  

Question: 38

27500000=2.75× 10 ? MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaaiE dacaaI1aGaaGimaiaaicdacaaIWaGaaGimaiaaicdacqGH9aqpcaaI YaGaaiOlaiaaiEdacaaI1aGaey41aqRaaGymaiaaicdadaahaaWcbe qaaiaac+daaaaaaa@4427@

Solution

The given number =   27500000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIYaGaaG4naiaaiwdacaaIWaGaaGimaiaaicdacaaIWaGaaGim aaaa@3BD6@

This number can be expressed in standard form as

27500000 =275× 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaqaaaaa aaaaWdbiaaikdacaaI3aGaaGynaiaaicdacaaIWaGaaGimaiaaicda caaIWaaabaGaeyypa0JaaGOmaiaaiEdacaaI1aGaey41aqRaaGymai aaicdadaahaaWcbeqaaiaaiwdaaaaaaaa@4397@  

Also,

275 =2.75× 10 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaqaaaaa aaaaWdbiaaikdacaaI3aGaaGynaaqaaiabg2da9iaaikdacaGGUaGa aG4naiaaiwdacqGHxdaTcaaIXaGaaGimamaaCaaaleqabaGaaGOmaa aaaaaa@40A4@  

So,

2.75× 10 2 × 10 5 =2.75× 10 7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaaqaaaaa aaaaWdbiaaikdacaGGUaGaaG4naiaaiwdacqGHxdaTcaaIXaGaaGim amaaCaaaleqabaGaaGOmaaaakiabgEna0kaaigdacaaIWaWaaWbaaS qabeaacaaI1aaaaaGcbaGaeyypa0JaaGOmaiaac6cacaaI3aGaaGyn aiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaI3aaaaaaaaa@4A5B@                     [ a m × a n = a m+n ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaqa aaaaaaaaWdbiablwJirjaadggadaahaaWcbeqaaiaad2gaaaGccqGH xdaTcaWGHbWaaWbaaSqabeaacaWGUbaaaOGaeyypa0JaamyyamaaCa aaleqabaGaamyBaiabgUcaRiaad6gaaaaak8aacaGLBbGaayzxaaaa aa@4452@  

27500000 =2.75× 10 7  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaqaaaaa aaaaWdbiaaikdacaaI3aGaaGynaiaaicdacaaIWaGaaGimaiaaicda caaIWaaabaGaeyypa0JaaGOmaiaac6cacaaI3aGaaGynaiabgEna0k aaigdacaaIWaWaaWbaaSqabeaacaaI3aGaaiiOaaaaaaaa@456F@  

Question: 39

340900000=3.409× 10 ? MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaais dacaaIWaGaaGyoaiaaicdacaaIWaGaaGimaiaaicdacaaIWaGaeyyp a0JaaG4maiaac6cacaaI0aGaaGimaiaaiMdacqGHxdaTcaaIXaGaaG imamaaCaaaleqabaGaai4paaaaaaa@459F@

Solution

The given number can be written in standard form as,

340900000 =3409× 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaqaaaaa aaaaWdbiaaiodacaaI0aGaaGimaiaaiMdacaaIWaGaaGimaiaaicda caaIWaGaaGimaaqaaiabg2da9iaaiodacaaI0aGaaGimaiaaiMdacq GHxdaTcaaIXaGaaGimamaaCaaaleqabaGaaGynaaaaaaaa@450F@  

Also,

3409 =3.409× 10 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaqaaaaa aaaaWdbiaaiodacaaI0aGaaGimaiaaiMdaaeaacqGH9aqpcaaIZaGa aiOlaiaaisdacaaIWaGaaGyoaiabgEna0kaaigdacaaIWaWaaWbaaS qabeaacaaIZaaaaaaaaa@421D@  

So,

3.409× 10 3 × 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIZaGaaiOlaiaaisdacaaIWaGaaGyoaiabgEna0kaaigdacaaI WaWaaWbaaSqabeaacaaIZaaaaOGaey41aqRaaGymaiaaicdadaahaa Wcbeqaaiaaiwdaaaaaaa@429A@                          [       a m × a n = a m+n ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaqa aaaaaaaaWdbiablwJirjaabccacaqGGaGaaeiiaiaabccacaqGGaGa amyyamaaCaaaleqabaGaamyBaaaakiabgEna0kaadggadaahaaWcbe qaaiaad6gaaaGccqGH9aqpcaWGHbWaaWbaaSqabeaacaWGTbGaey4k aSIaamOBaaaaaOWdaiaawUfacaGLDbaaaaa@4781@  

=3.409× 10 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGH9aqpcaaIZaGaaiOlaiaaisdacaaIWaGaaGyoaiabgEna0kaa igdacaaIWaWaaWbaaSqabeaacaaI4aaaaaaa@3F23@

340900000 =3.409× 10 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaaqaaaaa aaaaWdbiaaiodacaaI0aGaaGimaiaaiMdacaaIWaGaaGimaiaaicda caaIWaGaaGimaaqaaiabg2da9iaaiodacaGGUaGaaGinaiaaicdaca aI5aGaey41aqRaaGymaiaaicdadaahaaWcbeqaaiaaiIdaaaaaaaa@45C3@  

Question: 40

Fill in the blanks with <, > or = sign.

a.    3 2 ______15 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaCa aaleqabaGaaGOmaaaakiaac+facaGGFbGaai4xaiaac+facaGGFbGa ai4xaiaaykW7caaIXaGaaGynaaaa@3FDF@

b.   2 3 ______ 3 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaG4maaaakiaac+facaGGFbGaai4xaiaac+facaGGFbGa ai4xaiaaiodadaahaaWcbeqaaiaaikdaaaaaaa@3E80@

c.    7 4 ______ 5 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4namaaCa aaleqabaGaaGinaaaakiaac+facaGGFbGaai4xaiaac+facaGGFbGa ai4xaiaaykW7caaI1aWaaWbaaSqabeaacaaI0aaaaaaa@4015@

d.   10,000______ 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic dacaGGSaGaaGimaiaaicdacaaIWaGaai4xaiaac+facaGGFbGaai4x aiaac+facaGGFbGaaGymaiaaicdadaahaaWcbeqaaiaaiwdaaaaaaa@41DE@

e.    6 3  _____ 4 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnamaaCa aaleqabaGaaG4maaaakiaabccacaGGFbGaai4xaiaac+facaGGFbGa ai4xaiaaisdadaahaaWcbeqaaiaaisdaaaaaaa@3E47@

Solution

a.    3 2 ____15 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIZaWaaWbaaSqabeaacaaIYaaaaOGaai4xaiaac+facaGGFbGa ai4xaiaaigdacaaI1aaaaa@3CAE@  
We can write, 3 2  =3×3=9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIZaWaaWbaaSqabeaacaaIYaaaaOGaaiiOaiabg2da9iaaioda cqGHxdaTcaaIZaGaeyypa0JaaGyoaaaa@3F2C@  
 
So, 9<15 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaI5aGaeyipaWJaaGymaiaaiwdaaaa@3939@  
Therefore, 3 2 <15 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIZaWaaWbaaSqabeaacaaIYaaaaOGaeyipaWJaaGymaiaaiwda aaa@3A26@  

b.   2 3 ____ 3 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIYaWaaWbaaSqabeaacaaIZaaaaOGaai4xaiaac+facaGGFbGa ai4xaiaaiodadaahaaWcbeqaaiaaikdaaaaaaa@3CDA@  
We can write,
2 3  =2×2×2=8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIYaWaaWbaaSqabeaacaaIZaaaaOGaaiiOaiabg2da9iaaikda cqGHxdaTcaaIYaGaey41aqRaaGOmaiabg2da9iaaiIdaaaa@41FC@  
And 3 2 =3×3=9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIZaWaaWbaaSqabeaacaaIYaaaaOGaeyypa0JaaG4maiabgEna 0kaaiodacqGH9aqpcaaI5aaaaa@3E08@  
So,
8<9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaI4aGaeyipaWJaaGyoaaaa@3881@  
Therefore, 2 3 < 3 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIYaWaaWbaaSqabeaacaaIZaaaaOGaeyipaWJaaG4mamaaCaaa leqabaGaaGOmaaaaaaa@3A52@  

c.    7 4 ____ 5 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaI3aWaaWbaaSqabeaacaaI0aaaaOGaai4xaiaac+facaGGFbGa ai4xaiaaiwdadaahaaWcbeqaaiaaisdaaaaaaa@3CE4@  
In the above expression, as base 7 is greater than base 5 and power is same, Therefore,
7 4 > 5 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaI3aWaaWbaaSqabeaacaaI0aaaaOGaeyOpa4JaaGynamaaCaaa leqabaGaaGinaaaaaaa@3A60@  

d.   10000__ 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIXaGaaGimaiaaicdacaaIWaGaaGimaiaac+facaGGFbGaaGym aiaaicdadaahaaWcbeqaaiaaiwdaaaaaaa@3DC2@  
We can write,
10000= 10 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIXaGaaGimaiaaicdacaaIWaGaaGimaiabg2da9iaaigdacaaI WaWaaWbaaSqabeaacaaI0aaaaaaa@3D01@   
So, 10 4 < 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIXaGaaGimamaaCaaaleqabaGaaGinaaaakiabgYda8iaaigda caaIWaWaaWbaaSqabeaacaaI1aaaaaaa@3BC7@  
Therefore, 10000< 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIXaGaaGimaiaaicdacaaIWaGaaGimaiabgYda8iaaigdacaaI WaWaaWbaaSqabeaacaaI1aaaaaaa@3D00@  

e.    6 3 ___ 4 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaI2aWaaWbaaSqabeaacaaIZaaaaOGaai4xaiaac+facaGGFbGa aGinamaaCaaaleqabaGaaGinaaaaaaa@3BFE@  
We can write,
6 3 =6×6×6=216 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaI2aWaaWbaaSqabeaacaaIZaaaaOGaeyypa0JaaGOnaiabgEna 0kaaiAdacqGHxdaTcaaI2aGaeyypa0JaaGOmaiaaigdacaaI2aaaaa@425D@  
And 4 4  =4×4×4×4=256 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaI0aWaaWbaaSqabeaacaaI0aaaaOGaaiiOaiabg2da9iaaisda cqGHxdaTcaaI0aGaey41aqRaaGinaiabgEna0kaaisdacqGH9aqpca aIYaGaaGynaiaaiAdaaaa@4653@
So,
216<256 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIYaGaaGymaiaaiAdacqGH8aapcaaIYaGaaGynaiaaiAdaaaa@3B6E@  
Therefore, 6 3 < 4 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaI2aWaaWbaaSqabeaacaaIZaaaaOGaeyipaWJaaGinamaaCaaa leqabaGaaGinaaaaaaa@3A59@  

 

In questions 41 to 65, state whether the given statements are True or False.

Question: 41

One million = 10 7 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ymaiaaicdadaahaaWcbeqaaiaaiEdaaaaaaa@3942@

Solution

False

We know that, One million=10 lakhs=1000000= 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGpbGaaeOBaiaabwgacaqGGaGaaeyBaiaabMgacaqGSbGaaeiB aiaabMgacaqGVbGaaeOBaiabg2da9iaaigdacaaIWaGaaeiiaiaabY gacaqGHbGaae4AaiaabIgacaqGZbGaeyypa0JaaGymaiaaicdacaaI WaGaaGimaiaaicdacaaIWaGaaGimaiabg2da9iaaigdacaaIWaWaaW baaSqabeaacaaI2aaaaaaa@5114@  

Hence, 10 6   10 7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIXaGaaGimamaaCaaaleqabaGaaGOnaaaakiaacckacqGHGjsU caaIXaGaaGimamaaCaaaleqabaGaaG4naaaaaaa@3DB2@  

Question: 42

One hour = 60 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG OnaiaaicdadaahaaWcbeqaaiaaikdaaaaaaa@3942@  seconds

Solution

True

We know that, 1 h=60 min=60×60 s= 60 2  s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIXaGaaeiiaiaabIgacqGH9aqpcaaI2aGaaGimaiaabccacaqG TbGaaeyAaiaab6gacqGH9aqpcaaI2aGaaGimaiabgEna0kaaiAdaca aIWaGaaeiiaiaabohacqGH9aqpcaaI2aGaaGimamaaCaaaleqabaGa aGOmaaaakiaabccacaqGZbaaaa@4AE7@  

Question: 43

1 0 × 0 1  =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIXaWaaWbaaSqabeaacaaIWaaaaOGaey41aqRaaGimamaaCaaa leqabaGaaGymaaaakiaacckacqGH9aqpcaaIXaaaaa@3E4C@

Solution

False

We know that, 1 0 × 0 1  =1×0=01 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIXaWaaWbaaSqabeaacaaIWaaaaOGaey41aqRaaGimamaaCaaa leqabaGaaGymaaaakiaacckacqGH9aqpcaaIXaGaey41aqRaaGimai abg2da9iaaicdacqGHGjsUcaaIXaaaaa@455F@

Question: 44

( 3 ) 4  =12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaqa aaaaaaaaWdbiabgkHiTiaaiodaa8aacaGLOaGaayzkaaWdbmaaCaaa leqabaGaaGinaaaakiaacckacqGH9aqpcqGHsislcaaIXaGaaGOmaa aa@3ECD@

Solution

False

We know that, ( 3 ) 4  =( 3 )×( 3 )×( 3 )×( 3 )=8112 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaqa aaaaaaaaWdbiabgkHiTiaaiodaa8aacaGLOaGaayzkaaWdbmaaCaaa leqabaGaaGinaaaakiaacckacqGH9aqppaWaaeWaaeaapeGaeyOeI0 IaaG4maaWdaiaawIcacaGLPaaapeGaey41aq7damaabmaabaWdbiab gkHiTiaaiodaa8aacaGLOaGaayzkaaWdbiabgEna0+aadaqadaqaa8 qacqGHsislcaaIZaaapaGaayjkaiaawMcaa8qacqGHxdaTpaWaaeWa aeaapeGaeyOeI0IaaG4maaWdaiaawIcacaGLPaaapeGaeyypa0JaaG ioaiaaigdacqGHGjsUcqGHsislcaaIXaGaaGOmaaaa@5720@  

Question: 45

3 4 > 4 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaCa aaleqabaGaaGinaaaakiabg6da+iaaisdadaahaaWcbeqaaiaaioda aaaaaa@3A3A@

Solution

True

We know that, 3 4 =3×3×3×3=81 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSynIeLaaG 4mamaaCaaaleqabaGaaGinaaaakiabg2da9iaaiodacqGHxdaTcaaI ZaGaey41aqRaaG4maiabgEna0kaaiodacqGH9aqpcaaI4aGaaGymaa aa@4585@  and

4 3  =4×4×4=64 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinamaaCa aaleqabaGaaG4maaaakiaacckacqGH9aqpcaaI0aGaey41aqRaaGin aiabgEna0kaaisdacqGH9aqpcaaI2aGaaGinaaaa@42A0@

81>64 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaaig dacqGH+aGpcaaI2aGaaGinaaaa@39DB@

Therefore, 3 4  > 4 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4mamaaCa aaleqabaGaaeinaiaacckaaaGccqGH+aGpcaqG0aWaaWbaaSqabeaa caqGZaaaaaaa@3B42@

Question: 46

( 3 5 ) 100 = 3 100 5 100 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq GHsisldaWcaaqaaiaaiodaaeaacaaI1aaaaaGaayjkaiaawMcaamaa CaaaleqabaGaaGymaiaaicdacaaIWaaaaOGaeyypa0ZaaSaaaeaacq GHsislcaaIZaWaaWbaaSqabeaacaaIXaGaaGimaiaaicdaaaaakeaa cqGHsislcaaI1aWaaWbaaSqabeaacaaIXaGaaGimaiaaicdaaaaaaa aa@456F@

Solution

True

Considering left hand side of the equation, we have

( 3 5 ) 100  = ( 1×3 5 ) 100 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq GHsisldaWcaaqaaiaaiodaaeaacaaI1aaaaaGaayjkaiaawMcaamaa CaaaleqabaGaaGymaiaaicdacaaIWaaaaOGaaiiOaiabg2da9maabm aabaWaaSaaaeaacqGHsislcaaIXaGaey41aqRaaG4maaqaaiaaiwda aaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIXaGaaGimaiaaicdaaa aaaa@479A@

= ( 1 ) 100 × 3 100 5 100 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS aaaeaadaqadaqaaiabgkHiTiaaigdaaiaawIcacaGLPaaadaahaaWc beqaaiaaigdacaaIWaGaaGimaaaakiabgEna0kaaiodadaahaaWcbe qaaiaaigdacaaIWaGaaGimaaaaaOqaaiaaiwdadaahaaWcbeqaaiaa igdacaaIWaGaaGimaaaaaaaaaa@44DA@

= 1× 3 100 5 100 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS aaaeaacaaIXaGaey41aqRaaG4mamaaCaaaleqabaGaaGymaiaaicda caaIWaaaaaGcbaGaaGynamaaCaaaleqabaGaaGymaiaaicdacaaIWa aaaaaaaaa@3FFE@

= 3 100 5 100 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS aaaeaacaaIZaWaaWbaaSqabeaacaaIXaGaaGimaiaaicdaaaaakeaa caaI1aWaaWbaaSqabeaacaaIXaGaaGimaiaaicdaaaaaaaaa@3D2C@

Now, taking RHS, we have 3 100 5 100  = 3 100 5 100   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIZaWaaWbaaSqabeaacaaIXaGaaGimaiaaicdaaaaakeaa cqGHsislcaaI1aWaaWbaaSqabeaacaaIXaGaaGimaiaaicdaaaaaaO GaaiiOaiabg2da9maalaaabaGaaG4mamaaCaaaleqabaGaaGymaiaa icdacaaIWaaaaaGcbaGaaGynamaaCaaaleqabaGaaGymaiaaicdaca aIWaaaaaaakiaacckaaaa@47B0@

LHS = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0daaa@36DF@  RHS

Hence, ( 3 5 ) 100 = 3 100 5 100 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq GHsisldaWcaaqaaiaaiodaaeaacaaI1aaaaaGaayjkaiaawMcaamaa CaaaleqabaGaaGymaiaaicdacaaIWaaaaOGaeyypa0ZaaSaaaeaacq GHsislcaaIZaWaaWbaaSqabeaacaaIXaGaaGimaiaaicdaaaaakeaa cqGHsislcaaI1aWaaWbaaSqabeaacaaIXaGaaGimaiaaicdaaaaaaa aa@456F@

Question: 47

( 10+10 ) 10 = 10 10 + 10 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIXaGaaGimaiabgUcaRiaaigdacaaIWaaacaGLOaGaayzkaaWaaWba aSqabeaacaaIXaGaaGimaaaakiabg2da9iaaigdacaaIWaWaaWbaaS qabeaacaaIXaGaaGimaaaakiabgUcaRiaaigdacaaIWaWaaWbaaSqa beaacaaIXaGaaGimaaaaaaa@44FA@

Solution

False

Let us solve LHS,

( 10+10 ) 10 = (20) 10 = (2×10) 10 = 2 10 × 10 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIXaGaaGimaiabgUcaRiaaigdacaaIWaaacaGLOaGaayzkaaWaaWba aSqabeaacaaIXaGaaGimaaaakiabg2da9iaacIcacaaIYaGaaGimai aacMcadaahaaWcbeqaaiaaigdacaaIWaaaaOGaeyypa0Jaaiikaiaa ikdacqGHxdaTcaaIXaGaaGimaiaacMcadaahaaWcbeqaaiaaigdaca aIWaaaaOGaeyypa0JaaGOmamaaCaaaleqabaGaaGymaiaaicdaaaGc cqGHxdaTcaaIXaGaaGimamaaCaaaleqabaGaaGymaiaaicdaaaaaaa@534A@

Let us solve RHS,

10 10 + 10 10 =2× 10 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic dadaahaaWcbeqaaiaaigdacaaIWaaaaOGaey4kaSIaaGymaiaaicda daahaaWcbeqaaiaaigdacaaIWaaaaOGaeyypa0JaaGOmaiabgEna0k aaigdacaaIWaWaaWbaaSqabeaacaaIXaGaaGimaaaaaaa@43ED@

Now,

2 10 × 10 10 2× 10 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaGymaiaaicdaaaGccqGHxdaTcaaIXaGaaGimamaaCaaa leqabaGaaGymaiaaicdaaaGccqGHGjsUcaaIYaGaey41aqRaaGymai aaicdadaahaaWcbeqaaiaaigdacaaIWaaaaaaa@452A@

Hence, LHS is not equal to RHS

 

Question: 48

x 0 ×  x 0 = x 0 ÷ x 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGimaaaakiabgEna0kaabccacaWG4bWaaWbaaSqabeaa caaIWaaaaOGaeyypa0JaamiEamaaCaaaleqabaGaaGimaaaakiabgE pa4kaadIhadaahaaWcbeqaaiaaicdaaaaaaa@4381@  is true for all non-zero values of x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D6@ .

Solution

True

As we know that

x 0 × x 0 =1×1=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGimaaaakiabgEna0kaadIhadaahaaWcbeqaaiaaicda aaGccqGH9aqpcaaIXaGaey41aqRaaGymaiabg2da9iaaigdaaaa@421F@                   [ a 0 =1] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiablw JirjaadggadaahaaWcbeqaaiaaicdaaaGccqGH9aqpcaaIXaGaaiyx aaaa@3C69@  and

x 0 ÷ x 0 =1÷1=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGimaaaakiabgEpa4kaadIhadaahaaWcbeqaaiaaicda aaGccqGH9aqpcaaIXaGaey49aGRaaGymaiabg2da9iaaigdaaaa@4267@                  [ a 0 =1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacq WI1isucaWGHbWaaWbaaSqabeaacaaIWaaaaOGaeyypa0JaaGymaaGa ay5waiaaw2faaaaa@3C9B@

Hence,   x 0 × x 0 = x 0 ÷ x 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiOaiaadI hadaahaaWcbeqaaiaaicdaaaGccqGHxdaTcaWG4bWaaWbaaSqabeaa caaIWaaaaOGaeyypa0JaamiEamaaCaaaleqabaGaaGimaaaakiabgE pa4kaadIhadaahaaWcbeqaaiaaicdaaaaaaa@4402@

Question: 49

In the standard form, a large number can be expressed as a decimal number between 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@3693@  and 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaaaa@3694@ , multiplied by a power of 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic daaaa@374E@ .

Solution

False

We know that, a number in standard form is written as a×  10 k , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgE na0kaabccacaaIXaGaaGimamaaCaaaleqabaGaam4AaaaakiaacYca aaa@3CC4@  where 1a<10 and k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgs MiJkaadggacqGH8aapcaaIXaGaaGimaiaabccacaqGHbGaaeOBaiaa bsgacaqGGaGaam4Aaaaa@4099@  is any integer.

Question: 50

4 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinamaaCa aaleqabaGaaGOmaaaaaaa@3780@  is greater than 2 4 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaGinaaaakiaac6caaaa@383C@  

Solution

False

4 2 =4×4=16 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinamaaCa aaleqabaGaaGOmaaaakiabg2da9iaaisdacqGHxdaTcaaI0aGaeyyp a0JaaGymaiaaiAdaaaa@3EA3@          [ a m =a×a×a××a (m times)] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiablw JirjaadggadaahaaWcbeqaaiaad2gaaaGccqGH9aqpcaWGHbGaey41 aqRaamyyaiabgEna0kaadggacqGHxdaTcqGHMacVcqGHxdaTcaWGHb GaaeiiaiaacIcacaWGTbGaaeiiaiaabshacaqGPbGaaeyBaiaabwga caqGZbGaaiykaiaac2faaaa@51AA@  and

2 4 =2×2×2×2=16 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaGinaaaakiabg2da9iaaikdacqGHxdaTcaaIYaGaey41 aqRaaGOmaiabgEna0kaaikdacqGH9aqpcaaIXaGaaGOnaaaa@4445@    

Therefore, 4 2  = 2 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinamaaCa aaleqabaGaaGOmaaaakiaacckacqGH9aqpcaaIYaWaaWbaaSqabeaa caaI0aaaaaaa@3B5A@

Question: 51

  x m + x m = x 2 m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaamyBaaaakiabgUcaRiaadIhadaahaaWcbeqaaiaad2ga aaGccqGH9aqpcaWG4bWaaWbaaSqabeaacaaIYaaaaOWaaWbaaSqabe aacaWGTbaaaaaa@3F1B@  where x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D6@  is a non-zero rational number and m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@36CB@  is a positive integer.

Solution

False

We have studied that

a m × a n = a m+n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaamyBaaaakiabgEna0kaadggadaahaaWcbeqaaiac0b4G UbaaaOGaeyypa0JaamyyamaaCaaaleqabaGaamyBaiabgUcaRiaad6 gaaaaaaa@41EA@

x m × x m = x m+m = x 2m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaamyBaaaakiabgEna0kaadIhadaahaaWcbeqaaiaad2ga aaGccqGH9aqpcaWG4bWaaWbaaSqabeaacaWGTbGaey4kaSIaamyBaa aakiabg2da9iaadIhadaahaaWcbeqaaiaaikdacaWGTbaaaaaa@4519@

Also, a k + a k =2  a k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaam4AaaaakiabgUcaRiaadggadaahaaWcbeqaaiaadUga aaGccqGH9aqpcaaIYaGaaeiiaiaadggadaahaaWcbeqaaiaadUgaaa aaaa@3F3C@

So, x m + x m =2 x m x 2m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaamyBaaaakiabgUcaRiaadIhadaahaaWcbeqaaiaad2ga aaGccqGH9aqpcaaIYaGaamiEamaaCaaaleqabaGaamyBaaaakiabgc Mi5kaadIhadaahaaWcbeqaaiaaikdacaWGTbaaaaaa@438D@

Question: 52

x m × y m = (x×y) 2 m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaamyBaaaakiabgEna0kaadMhadaahaaWcbeqaaiaad2ga aaGccqGH9aqpcaGGOaGaamiEaiabgEna0kaadMhacaGGPaWaaWbaaS qabeaacaaIYaaaaOWaaWbaaSqabeaacaWGTbaaaaaa@44BF@ , where x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D6@  and y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36D7@  are non-zero rational numbers and m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@36CB@  is a positive integer.

Solution

False

We have studied that if a and b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaabc cacaqGHbGaaeOBaiaabsgacaqGGaGaamOyaaaa@3BA7@  are rational numbers, then x m × y m = (xy) m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaamyBaaaakiabgEna0kaadMhadaahaaWcbeqaaiac0b4G TbaaaOGaeyypa0JaaiikaiaadIhacaWG5bGaaiykamaaCaaaleqaba GaamyBaaaaaaa@42B1@

x m × y m = (xy) m = (x×y) m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaamyBaaaakiabgEna0kaadMhadaahaaWcbeqaaiaad2ga aaGccqGH9aqpcaGGOaGaamiEaiaadMhacaGGPaWaaWbaaSqabeaaca WGTbaaaOGaeyypa0JaaiikaiaadIhacqGHxdaTcaWG5bGaaiykamaa CaaaleqabaGaamyBaaaaaaa@494F@

Hence, x m × y m (x×y) 2m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaamyBaaaakiabgEna0kaadMhadaahaaWcbeqaaiaad2ga aaGccqGHGjsUcaGGOaGaamiEaiabgEna0kaadMhacaGGPaWaaWbaaS qabeaacGaDaIOmaiac0b4GTbaaaaaa@4741@

Question: 53

x m ÷ y m = (x÷y) m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaamyBaaaakiabgEpa4kaadMhadaahaaWcbeqaaiaad2ga aaGccqGH9aqpcaGGOaGaamiEaiabgEpa4kaadMhacaGGPaWaaWbaaS qabeaacaWGTbaaaaaa@4414@  where x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D6@  and y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36D7@  are non-zero rational numbers and m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@36CB@  is a positive integer.

Solution

True

We have studied that, if x and y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaabc cacaqGHbGaaeOBaiaabsgacaqGGaGaamyEaaaa@3BD5@  are rational numbers, then

( x m y m ) = ( x y ) m =or  x m ÷ y m = (x÷y) m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaadIhadaahaaWcbeqaaiaad2gaaaaakeaacaWG5bWaaWba aSqabeaacaWGTbaaaaaaaOGaayjkaiaawMcaaiaabccacqGH9aqpda qadaqaamaalaaabaGaamiEaaqaaiaadMhaaaaacaGLOaGaayzkaaWa aWbaaSqabeaacaWGTbaaaOGaeyypa0Jaae4BaiaabkhacaqGGaGaam iEamaaCaaaleqabaGaamyBaaaakiabgEpa4kaadMhadaahaaWcbeqa aiaad2gaaaGccqGH9aqpcaGGOaGaamiEaiabgEpa4kaadMhacaGGPa WaaWbaaSqabeaacaWGTbaaaaaa@53F0@

Question: 54

x m × x n = x m +n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaamyBaaaakiabgEna0kaadIhadaahaaWcbeqaaiaad6ga aaGccqGH9aqpcaWG4bWaaWbaaSqabeaacaWGTbaaaOWaaWbaaSqabe aacqGHRaWkcaWGUbaaaaaa@416A@  where x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D6@  is a non-zero rational number and m,n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiaacY cacaWGUbaaaa@386E@  are positive integers.

Solution

True

We have studied that, if a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36BF@  is a rational number and m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@36CB@  and n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36CC@  are positive integers, then

a m × a n = a m+n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaamyBaaaakiabgEna0kaadggadaahaaWcbeqaaiaad6ga aaGccqGH9aqpcaWGHbWaaWbaaSqabeaacaWGTbGaey4kaSIaamOBaa aaaaa@40EE@

x m × x n = x m+n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIWLaam iEamaaCaaaleqabaGaamyBaaaakiabgEna0kaadIhadaahaaWcbeqa aiaad6gaaaGccqGH9aqpcaWG4bWaaWbaaSqabeaacaWGTbGaey4kaS IaamOBaaaaaaa@4271@

Question: 55

4 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinamaaCa aaleqabaGaaGyoaaaaaaa@3787@  is greater than 16 3 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaiA dadaahaaWcbeqaaiaaiodaaaGccaGGUaaaaa@38FA@

Solution

True

 16=4×4= 4 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSynIeLaae iiaiaaigdacaaI2aGaeyypa0JaaGinaiabgEna0kaaisdacqGH9aqp caaI0aWaaWbaaSqabeaacaaIYaaaaaaa@4075@                 16 3 = ( 4 2 ) 3 = 4 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIWLaae iiaiaaigdacaaI2aWaaWbaaSqabeaacaaIZaaaaOGaeyypa0ZaaeWa aeaacaaI0aWaaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaWaaW baaSqabeaacaaIZaaaaOGaeyypa0JaaGinamaaCaaaleqabaGaaGOn aaaaaaa@420D@            

Now, in 4 9  and  4 6 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinamaaCa aaleqabaGaaGyoaaaakiaacckacaqGHbGaaeOBaiaabsgacaqGGaGa aGinamaaCaaaleqabaGaaGOnaaaakiaacYcaaaa@3E78@ 4 9 > 4 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinamaaCa aaleqabaGaaGyoaaaakiabg6da+iaaisdadaahaaWcbeqaaiaaiAda aaaaaa@3A43@  as powers 9>6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoaiabg6 da+iaaiAdaaaa@3863@

Question: 56

( 2 5 ) 3 ÷ ( 5 2 ) 3 =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaikdaaeaacaaI1aaaaaGaayjkaiaawMcaamaaCaaaleqa baGaaG4maaaakiabgEpa4oaabmaabaWaaSaaaeaacaaI1aaabaGaaG OmaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiodaaaGccqGH9aqp caaIXaaaaa@41E5@

Solution

False

Taking LHS we get,      ( 2 5 ) 3 ÷ ( 5 2 ) 3 = ( 2 5 ) 3 × ( 2 5 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaikdaaeaacaaI1aaaaaGaayjkaiaawMcaamaaCaaaleqa baGaaG4maaaakiabgEpa4oaabmaabaWaaSaaaeaacaaI1aaabaGaaG OmaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiodaaaGccqGH9aqp daqadaqaamaalaaabaGaaGOmaaqaaiaaiwdaaaaacaGLOaGaayzkaa WaaWbaaSqabeaacaaIZaaaaOGaey41aq7aaeWaaeaadaWcaaqaaiaa ikdaaeaacaaI1aaaaaGaayjkaiaawMcaamaaCaaaleqabaGaaG4maa aaaaa@4B46@

= ( 2 5 ) 3+3 = ( 2 5 ) 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaadaWcaaqaaiaaikdaaeaacaaI1aaaaaGaayjkaiaawMcaamaa CaaaleqabaGaaG4maiabgUcaRiaaiodaaaGccqGH9aqpdaqadaqaam aalaaabaGaaGOmaaqaaiaaiwdaaaaacaGLOaGaayzkaaWaaWbaaSqa beaacaaI2aaaaaaa@418C@

Hence, ( 2 5 ) 3 ÷ ( 5 2 ) 3 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaikdaaeaacaaI1aaaaaGaayjkaiaawMcaamaaCaaaleqa baGaaG4maaaakiabgEpa4oaabmaabaWaaSaaaeaacaaI1aaabaGaaG OmaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiodaaaGccqGHGjsU caaIXaaaaa@42A5@

Question: 57

( 4 3 ) 5 × ( 5 7 ) 5 = ( 4 3 + 5 7 ) 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaisdaaeaacaaIZaaaaaGaayjkaiaawMcaamaaCaaaleqa baGaaGynaaaakiabgEna0oaabmaabaWaaSaaaeaacaaI1aaabaGaaG 4naaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiwdaaaGccqGH9aqp daqadaqaamaalaaabaGaaGinaaqaaiaaiodaaaGaey4kaSYaaSaaae aacaaI1aaabaGaaG4naaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaa iwdaaaaaaa@4780@

Solution

False

Taking LHS we get,      ( 4 3 ) 5 × ( 5 7 ) 5 = ( 4 3 × 5 7 ) 5 = ( 20 21 ) 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaisdaaeaacaaIZaaaaaGaayjkaiaawMcaamaaCaaaleqa baGaaGynaaaakiabgEna0oaabmaabaWaaSaaaeaacaaI1aaabaGaaG 4naaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiwdaaaGccqGH9aqp daqadaqaamaalaaabaGaaGinaaqaaiaaiodaaaGaey41aq7aaSaaae aacaaI1aaabaGaaG4naaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaa iwdaaaGccqGH9aqpdaqadaqaamaalaaabaGaaGOmaiaaicdaaeaaca aIYaGaaGymaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiwdaaaaa aa@4F37@

And      [ ( 4 3 )+( 5 7 ) ] 5 = ( 4 3 + 5 7 ) 5 = ( 43 21 ) 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada qadaqaamaalaaabaGaaGinaaqaaiaaiodaaaaacaGLOaGaayzkaaGa ey4kaSYaaeWaaeaadaWcaaqaaiaaiwdaaeaacaaI3aaaaaGaayjkai aawMcaaaGaay5waiaaw2faamaaCaaaleqabaGaiWiGiwdaaaGccqGH 9aqpdaqadaqaamaalaaabaGaaGinaaqaaiaaiodaaaGaey4kaSYaaS aaaeaacaaI1aaabaGaaG4naaaaaiaawIcacaGLPaaadaahaaWcbeqa aiaaiwdaaaGccqGH9aqpdaqadaqaamaalaaabaGaaGinaiaaiodaae aacaaIYaGaaGymaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiwda aaaaaa@4EE2@

As the base is not same for LHS value,

So, LHS MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiyIKlaaa@37A0@  RHS

Hence, ( 4 3 ) 5 × ( 5 7 ) 5 ( 4 3 + 5 7 ) 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaisdaaeaacaaIZaaaaaGaayjkaiaawMcaamaaCaaaleqa baGaaGynaaaakiabgEna0oaabmaabaWaaSaaaeaacaaI1aaabaGaaG 4naaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiwdaaaGccqGHGjsU daqadaqaamaalaaabaGaaGinaaqaaiaaiodaaaGaey4kaSYaaSaaae aacaaI1aaabaGaaG4naaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaa iwdaaaaaaa@4841@

Question: 58

( 5 8 ) 9 ÷ ( 5 8 ) 4 = ( 5 8 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaiwdaaeaacaaI4aaaaaGaayjkaiaawMcaamaaCaaaleqa baGaaGyoaaaakiabgEpa4oaabmaabaWaaSaaaeaacaaI1aaabaGaaG ioaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaisdaaaGccqGH9aqp daqadaqaamaalaaabaGaaGynaaqaaiaaiIdaaaaacaGLOaGaayzkaa WaaWbaaSqabeaacaaI0aaaaaaa@4541@

Solution

False

Taking LHS we get, ( 5 8 ) 9 ÷ ( 5 8 ) 4 = ( 5 8 ) 94 = ( 5 8 ) 5 ( 5 8 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaiwdaaeaacaaI4aaaaaGaayjkaiaawMcaamaaCaaaleqa baGaaGyoaaaakiabgEpa4oacWcyadaqaialGdGaSaUaaaeacWcOaia lGiwdaaeacWcOaialGiIdaaaaacGaSaAjkaiacWcOLPaaadGaSaYba aSqajalGbGaSakacWciI0aaaaOGaeyypa0ZaaeWaaeaadaWcaaqaai aaiwdaaeaacaaI4aaaaaGaayjkaiaawMcaamaaCaaaleqabaGaaGyo aiabgkHiTiaaisdaaaGccqGH9aqpdaqadaqaamaalaaabaGaaGynaa qaaiaaiIdaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI1aaaaOGa eyiyIK7aaeWaaeaadaWcaaqaaiaaiwdaaeaacaaI4aaaaaGaayjkai aawMcaamaaCaaaleqabaGaaGinaaaaaaa@61B5@

Hence, ( 5 8 ) 9 ÷ ( 5 8 ) 4 ( 5 8 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaiwdaaeaacaaI4aaaaaGaayjkaiaawMcaamaaCaaaleqa baGaaGyoaaaakiabgEpa4oaabmaabaWaaSaaaeaacaaI1aaabaGaaG ioaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaisdaaaGccqGHGjsU daqadaqaamaalaaabaGaaGynaaqaaiaaiIdaaaaacaGLOaGaayzkaa WaaWbaaSqabeaacaaI0aaaaaaa@4602@

Question: 59

( 7 3 ) 2 × ( 7 3 ) 5 = ( 7 3 ) 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaiEdaaeaacaaIZaaaaaGaayjkaiaawMcaamaaCaaaleqa baGaaGOmaaaakiabgEna0oaabmaabaWaaSaaaeaacaaI3aaabaGaaG 4maaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiwdaaaGccqGH9aqp daqadaqaamaalaaabaGaaG4naaqaaiaaiodaaaaacaGLOaGaayzkaa WaaWbaaSqabeaacaaIXaGaaGimaaaaaaa@45C6@

Solution

False

Taking LHS we get, ( 7 3 ) 2 × ( 7 3 ) 5 = ( 7 3 ) 2+5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaiEdaaeaacaaIZaaaaaGaayjkaiaawMcaamaaCaaaleqa baGaaGOmaaaakiabgEna0oaabmaabaWaaSaaaeaacaaI3aaabaGaaG 4maaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiwdaaaGccqGH9aqp daqadaqaamaalaaabaGaaG4naaqaaiaaiodaaaaacaGLOaGaayzkaa WaaWbaaSqabeaacaaIYaGaey4kaSIaaGynaaaaaaa@46AD@

= ( 7 3 ) 7 ( 7 3 ) 10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaadaWcaaqaaiaaiEdaaeaacaaIZaaaaaGaayjkaiaawMcaamaa CaaaleqabaGaaG4naaaakiabgcMi5oaabmaabaWaaSaaaeaacaaI3a aabaGaaG4maaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaigdacaaI Waaaaaaa@416D@

Hence, ( 7 3 ) 2 × ( 7 3 ) 5 ( 7 3 ) 10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaiEdaaeaacaaIZaaaaaGaayjkaiaawMcaamaaCaaaleqa baGaaGOmaaaakiabgEna0oaabmaabaWaaSaaaeaacaaI3aaabaGaaG 4maaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiwdaaaGccqGHGjsU daqadaqaamaalaaabaGaaG4naaqaaiaaiodaaaaacaGLOaGaayzkaa WaaWbaaSqabeaacaaIXaGaaGimaaaaaaa@4686@

Question: 60

5 0 × 25 0 × 125 0 = ( 5 0 ) 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynamaaCa aaleqabaGaaGimaaaakiabgEna0kaaikdacaaI1aWaaWbaaSqabeaa caaIWaaaaOGaey41aqRaaGymaiaaikdacaaI1aWaaWbaaSqabeaaca aIWaaaaOGaeyypa0JaaiikaiaaiwdadaahaaWcbeqaaiaaicdaaaGc caGGPaWaaWbaaSqabeaacaaI2aaaaaaa@4645@

Solution

True

Taking LHS we get,

5 0 × 25 0 × 125 0 =1×1×1( a 0 =1 ) =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaaI1a WaaWbaaSqabeaacaaIWaaaaOGaey41aqRaaGOmaiaaiwdadaahaaWc beqaaiaaicdaaaGccqGHxdaTcaaIXaGaaGOmaiaaiwdadaahaaWcbe qaaiaaicdaaaaakeaacqGH9aqpcaaIXaGaey41aqRaaGymaiabgEna 0kaaigdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8+aaeWaaeaacqWI1isucaWGHbWaaWbaaSqabeaacaaI WaaaaOGaeyypa0JaaGymaaGaayjkaiaawMcaaaqaaiabg2da9iaaig daaaaa@7764@

 

Now, taking RHS we get,

( 5 0 ) 6 = ( 1 ) 6 ( a 0 =1 ) =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaqada qaaiaaiwdadaahaaWcbeqaaiaaicdaaaaakiaawIcacaGLPaaadaah aaWcbeqaaiaaiAdaaaGccqGH9aqpcaaMc8+aaeWaaeaacaaIXaWaaW baaSqabeaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaiAdaaaGc caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVpaabmaabaGaeSynIeLaamyyamaaCaaaleqabaGaaGimaaaaki abg2da9iaaigdaaiaawIcacaGLPaaaaeaacqGH9aqpcaaIXaaaaaa@6268@

Hence, 5 0 × 25 0 × 125 0 = ( 5 0 ) 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynamaaCa aaleqabaGaaGimaaaakiabgEna0kaaikdacaaI1aWaaWbaaSqabeaa caaIWaaaaOGaey41aqRaaGymaiaaikdacaaI1aWaaWbaaSqabeaaca aIWaaaaOGaeyypa0ZaaeWaaeaacaaI1aWaaWbaaSqabeaacaaIWaaa aaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaI2aaaaaaa@4675@

Question: 61

876543=8× 10 5 +7× 10 4 +6× 10 3 +5× 10 2 +4× 10 1 +3× 10 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaaI4a GaaG4naiaaiAdacaaI1aGaaGinaiaaiodacqGH9aqpcaaI4aGaey41 aqRaaGymaiaaicdadaahaaWcbeqaaiaaiwdaaaGccqGHRaWkcaaI3a Gaey41aqRaaGymaiaaicdadaahaaWcbeqaaiaaisdaaaGccqGHRaWk caaI2aGaey41aqRaaGymaiaaicdadaahaaWcbeqaaiaaiodaaaGccq GHRaWkcaaI1aGaey41aqRaaGymaiaaicdadaahaaWcbeqaaiaaikda aaaakeaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7cqGHRaWkcaaI0aGaey41aqRaaGymaiaaicdadaahaaWcbeqa aiaaigdaaaGccqGHRaWkcaaIZaGaey41aqRaaGymaiaaicdadaahaa Wcbeqaaiaaicdaaaaaaaa@8444@

Solution

True

Considering the right hand side of the given equation, we get 8× 10 5 +7× 10 4 +6× 10 3 +5× 10 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiabgE na0kaaigdacaaIWaWaaWbaaSqabeaacaaI1aaaaOGaey4kaSIaaG4n aiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaI0aaaaOGaey4kaS IaaGOnaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaIZaaaaOGa ey4kaSIaaGynaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaIYa aaaaaa@4D78@   +4× 10 1 +3× 10 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4kaSIaaG inaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaIXaaaaOGaey4k aSIaaG4maiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaIWaaaaa aa@4208@

=8×100000+7×10000+6×1000+5×100 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ioaiabgEna0kaaigdacaaIWaGaaGimaiaaicdacaaIWaGaaGimaiab gUcaRiaaiEdacqGHxdaTcaaIXaGaaGimaiaaicdacaaIWaGaaGimai abgUcaRiaaiAdacqGHxdaTcaaIXaGaaGimaiaaicdacaaIWaGaey4k aSIaaGynaiabgEna0kaaigdacaaIWaGaaGimaaaa@51FA@

 +4×10+3×1                          [  a 0 =1] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiOaiabgU caRiaaysW7caaI0aGaey41aqRaaGymaiaaicdacqGHRaWkcaaIZaGa ey41aqRaaGymaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabccacaqGGaGaai4waiablwJirjaabccacaWGHbWa aWbaaSqabeaacaaIWaaaaOGaeyypa0JaaGymaiaac2faaaa@59E8@

=800000+70000+6000+500+40+3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ioaiaaicdacaaIWaGaaGimaiaaicdacaaIWaGaey4kaSIaaG4naiaa icdacaaIWaGaaGimaiaaicdacqGHRaWkcaaI2aGaaGimaiaaicdaca aIWaGaey4kaSIaaGynaiaaicdacaaIWaGaey4kaSIaaGinaiaaicda cqGHRaWkcaaIZaaaaa@4AAB@

=876543=LHS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ioaiaaiEdacaaI2aGaaGynaiaaisdacaaIZaGaeyypa0Jaaeitaiaa bIeacaqGtbaaaa@3ED1@

Hence, RHS = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0daaa@36DF@  LHS

Question: 62

600060=6× 10 5 +6× 10 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnaiaaic dacaaIWaGaaGimaiaaiAdacaaIWaGaeyypa0JaaGOnaiabgEna0kaa igdacaaIWaWaaWbaaSqabeaacaaI1aaaaOGaey4kaSIaaGOnaiabgE na0kaaigdacaaIWaWaaWbaaSqabeaacaaIYaaaaaaa@469F@

Solution

False

Considering the right hand side of the given equation, we get   =6× 10 5 +6× 10 2 =6×100000+6×100 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG OnaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaI1aaaaOGaey4k aSIaaGOnaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaIYaaaaO Gaeyypa0JaaGOnaiabgEna0kaaigdacaaIWaGaaGimaiaaicdacaaI WaGaaGimaiabgUcaRiaaiAdacqGHxdaTcaaIXaGaaGimaiaaicdaaa a@5063@

=600000+600=600600600060 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG OnaiaaicdacaaIWaGaaGimaiaaicdacaaIWaGaey4kaSIaaGOnaiaa icdacaaIWaGaeyypa0JaaGOnaiaaicdacaaIWaGaaGOnaiaaicdaca aIWaGaeyiyIKRaaGOnaiaaicdacaaIWaGaaGimaiaaiAdacaaIWaaa aa@49F3@

Hence, RHSLHS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOuaiaabI eacaqGtbGaeyiyIKRaaeitaiaabIeacaqGtbaaaa@3C85@

Question: 63

4× 10 5 +3× 10 4 +2× 10 3 +1× 10 0 =432010 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiabgE na0kaaigdacaaIWaWaaWbaaSqabeaacaaI1aaaaOGaey4kaSIaaG4m aiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaI0aaaaOGaey4kaS IaaGOmaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaIZaaaaOGa ey4kaSIaaGymaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaIWa aaaOGaeyypa0JaaGinaiaaiodacaaIYaGaaGimaiaaigdacaaIWaaa aa@52DC@

Solution

False

Considering the left hand side of the given equation, we get =4×1 0 5 +3×1 0 4 +2×1 0 3 +1×1 0 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaae inaiabgEna0kaabgdacaaIWaWaaWbaaSqabeaacaqG1aaaaOGaey4k aSIaae4maiabgEna0kaabgdacaaIWaWaaWbaaSqabeaacaqG0aaaaO Gaey4kaSIaaeOmaiabgEna0kaabgdacaaIWaWaaWbaaSqabeaacaqG ZaaaaOGaey4kaSIaaeymaiabgEna0kaabgdacaaIWaWaaWbaaSqabe aacaaIWaaaaaaa@4E1F@

=4×100000+3×10000+2×1000+1×1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG inaiabgEna0kaaigdacaaIWaGaaGimaiaaicdacaaIWaGaaGimaiab gUcaRiaaiodacqGHxdaTcaaIXaGaaGimaiaaicdacaaIWaGaaGimai abgUcaRiaaikdacqGHxdaTcaaIXaGaaGimaiaaicdacaaIWaGaey4k aSIaaGymaiabgEna0kaaigdaaaa@5076@    [  a 0 =1] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiablw JirjaabccacaWGHbWaaWbaaSqabeaacaaIWaaaaOGaeyypa0JaaGym aiaac2faaaa@3D0C@

=400000+30000+2000+1=432001432010 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG inaiaaicdacaaIWaGaaGimaiaaicdacaaIWaGaey4kaSIaaG4maiaa icdacaaIWaGaaGimaiaaicdacqGHRaWkcaaIYaGaaGimaiaaicdaca aIWaGaey4kaSIaaGymaiabg2da9iaaisdacaaIZaGaaGOmaiaaicda caaIWaGaaGymaiabgcMi5kaaisdacaaIZaGaaGOmaiaaicdacaaIXa GaaGimaaaa@50C7@

Hence, LHSRHS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeitaiaabI eacaqGtbGaeyiyIKRaaeOuaiaabIeacaqGtbaaaa@3C85@

Question: 64

8× 10 6 +2× 10 4 +5× 10 2 +9× 10 0 =8020509 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiabgE na0kaaigdacaaIWaWaaWbaaSqabeaacaaI2aaaaOGaey4kaSIaaGOm aiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaI0aaaaOGaey4kaS IaaGynaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaIYaaaaOGa ey4kaSIaaGyoaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaIWa aaaOGaeyypa0JaaGioaiaaicdacaaIYaGaaGimaiaaiwdacaaIWaGa aGyoaaaa@53B2@

Solution MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcaa@35D8@

True

Considering the left hand side of the given equation, we get =8× 10 6 +2× 10 4 +5× 10 2 +9× 10 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ioaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaI2aaaaOGaey4k aSIaaGOmaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaI0aaaaO Gaey4kaSIaaGynaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaI YaaaaOGaey4kaSIaaGyoaiabgEna0kaaigdacaaIWaWaaWbaaSqabe aacaaIWaaaaaaa@4E7A@

=8×1000000+2×10000+5×100+9×1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ioaiabgEna0kaaigdacaaIWaGaaGimaiaaicdacaaIWaGaaGimaiaa icdacqGHRaWkcaaIYaGaey41aqRaaGymaiaaicdacaaIWaGaaGimai aaicdacqGHRaWkcaaI1aGaey41aqRaaGymaiaaicdacaaIWaGaey4k aSIaaGyoaiabgEna0kaaigdaaaa@5084@    [a°=1] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiablw JirjaadggacqGHWcaScqGH9aqpcaaIXaGaaiyxaaaa@3D64@

=8000000+20000+500+9=8020509=RHS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ioaiaaicdacaaIWaGaaGimaiaaicdacaaIWaGaaGimaiabgUcaRiaa ikdacaaIWaGaaGimaiaaicdacaaIWaGaey4kaSIaaGynaiaaicdaca aIWaGaey4kaSIaaGyoaiabg2da9iaaiIdacaaIWaGaaGOmaiaaicda caaI1aGaaGimaiaaiMdacqGH9aqpcaqGsbGaaeisaiaabofaaaa@4EEC@

Hence,  LHS=RHS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiOaiaabY eacaqGibGaae4uaiabg2da9iaabkfacaqGibGaae4uaaaa@3CE8@

Question: 65

4 0 + 5 0 + 6 0 = (4+5+6) 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinamaaCa aaleqabaGaaGimaaaakiabgUcaRiaaiwdadaahaaWcbeqaaiaaicda aaGccqGHRaWkcaaI2aWaaWbaaSqabeaacaaIWaaaaOGaeyypa0Jaai ikaiaaisdacqGHRaWkcaaI1aGaey4kaSIaaGOnaiaacMcadaahaaWc beqaaiaaicdaaaaaaa@43F3@

Solution

False

Considering the left hand side of the given equation, we get, 4 0 + 5 0 + 6 0 =1+1+1=3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinamaaCa aaleqabaGaaGimaaaakiabgUcaRiaaiwdadaahaaWcbeqaaiaaicda aaGccqGHRaWkcaaI2aWaaWbaaSqabeaacaaIWaaaaOGaeyypa0JaaG ymaiabgUcaRiaaigdacqGHRaWkcaaIXaGaeyypa0JaaG4maaaa@436A@     [ a 0 =1] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiablw JirjaadggadaahaaWcbeqaaiaaicdaaaGccqGH9aqpcaaIXaGaaiyx aaaa@3C69@  
and ( 4+5+6 ) 0 = ( 15 ) 0 =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aI0aGaey4kaSIaaGynaiabgUcaRiaaiAdaaiaawIcacaGLPaaadaah aaWcbeqaaiaaicdaaaGccqGH9aqpdaqadaqaaiaaigdacaaI1aaaca GLOaGaayzkaaWaaWbaaSqabeaacaaIWaaaaOGaeyypa0JaaGymaaaa @430E@

Hence, 4 0 + 5 0 + 6 0 (4+5+6) 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinamaaCa aaleqabaGaaGimaaaakiabgUcaRiaaiwdadaahaaWcbeqaaiaaicda aaGccqGHRaWkcaaI2aWaaWbaaSqabeaacaaIWaaaaOGaeyiyIKRaai ikaiaaisdacqGHRaWkcaaI1aGaey4kaSIaaGOnaiaacMcadaahaaWc beqaaiaaicdaaaaaaa@44B4@

Question: 66

Arrange in ascending order:

2 5 ,  3 3 ,  2 3 ×2,  ( 3 3 ) 2 ,  3 5 ,  4 0 ,  2 3 × 3 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaGynaaaakiaacYcacaqGGaGaaG4mamaaCaaaleqabaGa aG4maaaakiaacYcacaqGGaGaaGOmamaaCaaaleqabaGaaG4maaaaki abgEna0kaaikdacaGGSaGaaeiiaiaacIcacaaIZaWaaWbaaSqabeaa caaIZaaaaOGaaiykamaaCaaaleqabaGaaGOmaaaakiaacYcacaqGGa GaaG4mamaaCaaaleqabaGaaGynaaaakiaacYcacaqGGaGaaGinamaa CaaaleqabaGaaGimaaaakiaacYcacaqGGaGaaGOmamaaCaaaleqaba GaaG4maaaakiabgEna0kaaiodadaahaaWcbeqaaiaaigdaaaaaaa@527B@

Solution

Ascending order means arranging the numbers from least to greatest.

We have, 2 5 =2×2×2×2×2=32. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaGynaaaakiabg2da9iaaikdacqGHxdaTcaaIYaGaey41 aqRaaGOmaiabgEna0kaaikdacqGHxdaTcaaIYaGaeyypa0JaaG4mai aaikdacaGGUaaaaa@47C9@

3 3 =3×3×3=27. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaCa aaleqabaGaaG4maaaakiabg2da9iaaiodacqGHxdaTcaaIZaGaey41 aqRaaG4maiabg2da9iaaikdacaaI3aGaaiOlaaaa@4229@

2 3 ×2=2×2×2×2=16. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaG4maaaakiabgEna0kaaikdacqGH9aqpcaaIYaGaey41 aqRaaGOmaiabgEna0kaaikdacqGHxdaTcaaIYaGaeyypa0JaaGymai aaiAdacaGGUaaaaa@47C9@

( 3 3 ) 2 = 3 6       [ ( a m ) n = a mn ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaio dadaahaaWcbeqaaiaaiodaaaGccaGGPaWaaWbaaSqabeaacaaIYaaa aOGaeyypa0JaaG4mamaaCaaaleqabaGaaGOnaaaakiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaacUfacqWI1isucaGGOaGaamyy amaaCaaaleqabaGaamyBaaaakiaacMcadaahaaWcbeqaaiaad6gaaa GccqGH9aqpcaWGHbWaaWbaaSqabeaacaWGTbGaamOBaaaakiaac2fa aaa@4BF4@

3 6 =3×3×3×3×3×3=729. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaCa aaleqabaGaaGOnaaaakiabg2da9iaaiodacqGHxdaTcaaIZaGaey41 aqRaaG4maiabgEna0kaaiodacqGHxdaTcaaIZaGaey41aqRaaG4mai abg2da9iaaiEdacaaIYaGaaGyoaiaac6caaaa@4B6B@

3 5 =3×3×3×3×3=243. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaCa aaleqabaGaaGynaaaakiabg2da9iaaiodacqGHxdaTcaaIZaGaey41 aqRaaG4maiabgEna0kaaiodacqGHxdaTcaaIZaGaeyypa0JaaGOmai aaisdacaaIZaGaaiOlaaaa@488D@

4°=1                  [ a 0 =1] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiabgc laWkabg2da9iaaigdacaGGGcGaaiiOaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaiiOaiaacckacaGGBbGaeSynIeLaamyyamaa CaaaleqabaGaaGimaaaakiabg2da9iaaigdacaGGDbaaaa@4E4E@

and 2 3 ×3=2×2×2×3=24 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaG4maaaakiabgEna0kaaiodacqGH9aqpcaaIYaGaey41 aqRaaGOmaiabgEna0kaaikdacqGHxdaTcaaIZaGaeyypa0JaaGOmai aaisdaaaa@4718@

Thus, the required ascending order will be

4 0 < 2 3  ×2< 2 3 × 3 1 < 3 3 < 2 5 < 3 5 < ( 3 3 ) 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinamaaCa aaleqabaGaaGimaaaakiabgYda8iaaikdadaahaaWcbeqaaiaaioda aaGccaGGGcGaey41aqRaaGOmaiabgYda8iaaikdadaahaaWcbeqaai aaiodaaaGccqGHxdaTcaaIZaWaaWbaaSqabeaacaaIXaaaaOGaeyip aWJaaG4mamaaCaaaleqabaGaaG4maaaakiabgYda8iaaikdadaahaa WcbeqaaiaaiwdaaaGccqGH8aapcaaIZaWaaWbaaSqabeaacaaI1aaa aOGaeyipaWZaaeWaaeaacaaIZaWaaWbaaSqabeaacaaIZaaaaaGcca GLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaaiOlaaaa@52B1@

Question: 67

Arrange in descending order:

2 2+3 ,  ( 2 2 ) 3 , 2× 2 2 ,  3 5 3 2  ,  3 2 × 3 0 ,  2 3 × 5 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaGOmaiabgUcaRiaaiodaaaGccaGGSaGaaeiiaiaacIca caaIYaWaaWbaaSqabeaacaaIYaaaaOGaaiykamaaCaaaleqabaGaaG 4maaaakiaacYcacaqGGaGaaGOmaiabgEna0kaaikdadaahaaWcbeqa aiaaikdaaaGccaGGSaGaaeiiamaalaaabaGaaG4mamaaCaaaleqaba GaaGynaaaaaOqaaiaaiodadaahaaWcbeqaaiaaikdaaaaaaOGaaeii aiaacYcacaqGGaGaaG4mamaaCaaaleqabaGaaGOmaaaakiabgEna0k aaiodadaahaaWcbeqaaiaaicdaaaGccaGGSaGaaeiiaiaaikdadaah aaWcbeqaaiaaiodaaaGccqGHxdaTcaaI1aWaaWbaaSqabeaacaaIYa aaaaaa@573D@

Solution

Descending order means arranging the numbers from greatest to least.

We have, 2 2+3 = 2 5 =2×2×2×2×2=32. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaGOmaiabgUcaRiaaiodaaaGccqGH9aqpcaaIYaWaaWba aSqabeaacaaI1aaaaOGaeyypa0JaaGOmaiabgEna0kaaikdacqGHxd aTcaaIYaGaey41aqRaaGOmaiabgEna0kaaikdacqGH9aqpcaaIZaGa aGOmaiaac6caaaa@4C1D@

( 2 2 ) 3 = 2 6 =2×2×2×2×2×2=64. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaik dadaahaaWcbeqaaiaaikdaaaGccaGGPaWaaWbaaSqabeaacaaIZaaa aOGaeyypa0JaaGOmamaaCaaaleqabaGaaGOnaaaakiabg2da9iaaik dacqGHxdaTcaaIYaGaey41aqRaaGOmaiabgEna0kaaikdacqGHxdaT caaIYaGaey41aqRaaGOmaiabg2da9iaaiAdacaaI0aGaaiOlaaaa@4FA4@

2× 2 2  = 2 1+2 = 2 3 =8. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiabgE na0kaaikdadaahaaWcbeqaaiaaikdacaGGGcaaaOGaeyypa0JaaGOm amaaCaaaleqabaGaaGymaiabgUcaRiaaikdaaaGccqGH9aqpcaaIYa WaaWbaaSqabeaacaaIZaaaaOGaeyypa0JaaGioaiaac6caaaa@4500@

3 5 3 2  = 3 52 = 3 3 =27. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa aGOmaaaaaaGccaGGGcGaeyypa0JaaG4mamaaCaaaleqabaGaaGynai abgkHiTiaaikdaaaGccqGH9aqpcaaIZaWaaWbaaSqabeaacaaIZaaa aOGaeyypa0JaaGOmaiaaiEdacaGGUaaaaa@44BD@

3 2 × 3 0 = 3 2+0 = 3 2 =9. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaCa aaleqabaGaiWiGikdaaaGccqGHxdaTcaaIZaWaaWbaaSqabeaacaaI WaaaaOGaeyypa0JaaG4mamaaCaaaleqabaGaaGOmaiabgUcaRiaaic daaaGccqGH9aqpcaaIZaWaaWbaaSqabeaacaaIYaaaaOGaeyypa0Ja aGyoaiaac6caaaa@45E4@

2 3 × 5 2 =2×2×2×5×5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaG4maaaakiabgEna0kaaiwdadaahaaWcbeqaaiacmciI YaaaaOGaeyypa0JaaGOmaiabgEna0kaaikdacqGHxdaTcaaIYaGaey 41aqRaaGynaiabgEna0kaaiwdaaaa@4979@

=8×25=200. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ioaiabgEna0kaaikdacaaI1aGaeyypa0JaaGOmaiaaicdacaaIWaGa aiOlaaaa@3F1A@

Thus, the required descending order will be

( 2 3 × 5 2 )> ( 2 2 ) 3 > 2 2+3 > 3 5 3 2 >( 3 2 × 3 0 )>(2× 2 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaik dadaahaaWcbeqaaiaaiodaaaGccqGHxdaTcaaI1aWaaWbaaSqabeaa caaIYaaaaOGaaiykaiabg6da+iaacIcacaaIYaWaaWbaaSqabeaaca aIYaaaaOGaaiykamaaCaaaleqabaGaaG4maaaakiabg6da+iaaikda daahaaWcbeqaaiaaikdacqGHRaWkcaaIZaaaaOGaeyOpa4ZaaSaaae aacaaIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqa baGaaGOmaaaaaaGccqGH+aGpcaGGOaGaaG4mamaaCaaaleqabaGaaG OmaaaakiabgEna0kaaiodadaahaaWcbeqaaiaaicdaaaGccaGGPaGa eyOpa4JaaiikaiaaikdacqGHxdaTcaaIYaWaaWbaaSqabeaacaaIYa aaaOGaaiykaaaa@5938@

Question: 68

By what number should ( 4 ) 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq GHsislcaaI0aaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI1aaaaaaa @39F9@  be divided so that the quotient may be equal to ( 4 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq GHsislcaaI0aaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaaaaaaa @39F7@ ?

Solution

In order to find the number, that should divide (  4 ) 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca GGtaIaaeiiaiaaisdaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiwda aaaaaa@3A65@  to get the quotient (  4 ) 3 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca GGtaIaaeiiaiaaisdaaiaawIcacaGLPaaadaahaaWcbeqaaiaaioda aaGccaGGSaaaaa@3B1D@  we will divide ( 4) 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaco bicaqGGaGaaGinaiaacMcadaahaaWcbeqaaiaaiwdaaaaaaa@3A35@  by (  4 ) 3 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca GGtaIaaeiiaiaaisdaaiaawIcacaGLPaaadaahaaWcbeqaaiaaioda aaGccaGGUaaaaa@3B1F@   = (  4 ) 53 = ( 4) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaacaGGtaIaaeiiaiaaisdaaiaawIcacaGLPaaadaahaaWcbeqa aiaaiwdacqGHsislcaaIZaaaaOGaeyypa0JaaiikaiabgkHiTiaabc cacaaI0aGaaiykamaaCaaaleqabaGaaGOmaaaaaaa@42B5@

Hence, the required number is ( 4) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabgk HiTiaabccacaaI0aGaaiykamaaCaaaleqabaGaaGOmaaaaaaa@3A68@

Question: 69

Find m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@36CB@  so that ( 2 9 ) 3 × ( 2 9 ) 6 = ( 2 9 ) 2m1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaikdaaeaacaaI5aaaaaGaayjkaiaawMcaamaaCaaaleqa baGaaG4maaaakiabgEna0oaabmaabaWaaSaaaeaacaaIYaaabaGaaG yoaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiAdaaaGccqGH9aqp daqadaqaamaalaaabaGaaGOmaaqaaiaaiMdaaaaacaGLOaGaayzkaa WaaWbaaSqabeaacaaIYaGaamyBaiabgkHiTiaaigdaaaaaaa@47AB@  

Solution

We have,

( 2 9 ) 3 × ( 2 9 ) 6 = ( 2 9 ) 2 m1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaikdaaeaacaaI5aaaaaGaayjkaiaawMcaamaaCaaaleqa baGaaG4maaaakiabgEna0oaabmaabaWaaSaaaeaacaaIYaaabaGaaG yoaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiAdaaaGccqGH9aqp daqadaqaamaalaaabaGaaGOmaaqaaiaaiMdaaaaacaGLOaGaayzkaa WaaWbaaSqabeaacaaIYaGaaeiiaiaad2gacqGHsislcaaIXaaaaaaa @484E@

( 2 9 ) 3+6 = ( 2 9 ) 2m1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H49aae WaaeaadaWcaaqaaiaaikdaaeaacaaI5aaaaaGaayjkaiaawMcaamaa CaaaleqabaGaaG4maiabgUcaRiaaiAdaaaGccqGH9aqpdaqadaqaam aalaaabaGaaGOmaaqaaiaaiMdaaaaacaGLOaGaayzkaaWaaWbaaSqa beaacaaIYaGaamyBaiabgkHiTiaaigdaaaaaaa@4584@

( 2 9 ) 9 = ( 2 9 ) 2m1                a m × a n = a m+n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H49aae WaaeaadaWcaaqaaiaaikdaaeaacaaI5aaaaaGaayjkaiaawMcaamaa CaaaleqabaGaaGyoaaaakiabg2da9maabmaabaWaaSaaaeaacaaIYa aabaGaaGyoaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdacaWG TbGaeyOeI0IaaGymaaaakiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caqGGaGaeSynIeLaamyyamaaCaaaleqabaGaamyBaaaakiabgEna0k aadggadaahaaWcbeqaaiaad6gaaaGccqGH9aqpcaWGHbWaaWbaaSqa beaacaWGTbGaey4kaSIaamOBaaaaaaa@592B@

When the bases are same, we can equate the powers.

Therefore, 9=2m1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoaiabg2 da9iaaikdacaWGTbGaeyOeI0IaaGymaaaa@3AF7@

9+1=2m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoaiabgU caRiaaigdacqGH9aqpcaaIYaGaamyBaaaa@3AEC@

10=2m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic dacqGH9aqpcaaIYaGaamyBaaaa@3A01@

10 2 = 2m 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaGaaGimaaqaaiaaikdaaaGaeyypa0ZaaSaaaeaacaaIYaGaamyB aaqaaiaaikdaaaaaaa@3B99@

5=m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaiabg2 da9iaad2gaaaa@388F@

Hence, m=5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabg2 da9iaaiwdaaaa@388F@

Question: 70

If p q = ( 3 2 ) 2 ÷ ( 9 4 ) 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGWbaabaGaamyCaaaacqGH9aqpdaqadaqaamaalaaabaGaaG4maaqa aiaaikdaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaey 49aG7aaeWaaeaadaWcaaqaaiaaiMdaaeaacaaI0aaaaaGaayjkaiaa wMcaamaaCaaaleqabaGaaGimaaaaaaa@431B@  find the value of ( p q ) 3 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaadchaaeaacaWGXbaaaaGaayjkaiaawMcaamaaCaaaleqa baGaaG4maaaakiaac6caaaa@3B03@

Solution

Considering the given equation, p q = ( 3 2 ) 2 ÷ ( 9 4 ) 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGWbaabaGaamyCaaaacqGH9aqpdaqadaqaamaalaaabaGaaG4maaqa aiaaikdaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaey 49aG7aaeWaaeaadaWcaaqaaiaaiMdaaeaacaaI0aaaaaGaayjkaiaa wMcaamaaCaaaleqabaGaaGimaaaaaaa@431A@

p q = ( 3 2 ) 2 ÷ 1 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGWbaabaGaamyCaaaacqGH9aqpdaqadaqaamaalaaabaGaaG4maaqa aiaaikdaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaey 49aG7aaSaaaeaacaaIXaaabaGaaGymaaaaaaa@409F@

p q = ( 3 2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGWbaabaGaamyCaaaacqGH9aqpdaqadaqaamaalaaabaGaaG4maaqa aiaaikdaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaa@3CD4@

p q = 3 2 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGWbaabaGaamyCaaaacqGH9aqpdaWcaaqaaiaaiodadaahaaWcbeqa aiaaikdaaaaakeaacaaIYaWaaWbaaSqabeaacGaG0HOmaaaaaaaaaa@3D3A@

p q = 9 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGWbaabaGaamyCaaaacqGH9aqpdaWcaaqaaiaaiMdaaeaacaaI0aaa aaaa@3A6A@

Cubing both sides, we get

( p q ) 3 = ( 9 4 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaadchaaeaacaWGXbaaaaGaayjkaiaawMcaamaaCaaaleqa baGaaG4maaaakiabg2da9maabmaabaWaaSaaaeaacaaI5aaabaGaaG inaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiodaaaaaaa@3F5A@

( p q ) 3 = 9×9×9 4×4×4 = 729 64 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaqada qaamaalaaabaGaamiCaaqaaiaadghaaaaacaGLOaGaayzkaaWaaWba aSqabeaacaaIZaaaaOGaeyypa0ZaaSaaaeaacaaI5aGamWiGgEna0k aaiMdacqGHxdaTcaaI5aaabaGaaGinaiabgEna0kaaisdacqGHxdaT caaI0aaaaaqaaiabg2da9maalaaabaGaaG4naiaaikdacaaI5aaaba GaaGOnaiaaisdaaaaaaaa@4E33@

Question: 71

Find the reciprocal of the rational number ( 1 2 ) 2 ÷ ( 2 3 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaigdaaeaacaaIYaaaaaGaayjkaiaawMcaamaaCaaaleqa baGaaGOmaaaakiabgEpa4oaabmaabaWaaSaaaeaacaaIYaaabaGaaG 4maaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiodaaaaaaa@4013@

Solution

Considering the given expression

( 1 2 ) 2 ÷ ( 2 3 ) 3 = ( 1 2 ) 2 ( 2 3 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaqada qaamaalaaabaGaaGymaaqaaiaaikdaaaaacaGLOaGaayzkaaWaaWba aSqabeaacaaIYaaaaOGaey49aG7aaeWaaeaadaWcaaqaaiaaikdaae aacaaIZaaaaaGaayjkaiaawMcaamaaCaaaleqabaGaaG4maaaaaOqa aiabg2da9maalaaabaWaaeWaaeaadaWcaaqaaiaaigdaaeaacaaIYa aaaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaOqaamaabmaa baWaaSaaaeaacaaIYaaabaGaaG4maaaaaiaawIcacaGLPaaadaahaa Wcbeqaaiaaiodaaaaaaaaaaa@4937@

= (1) 2 (2) 2 (2) 3 (3) 3 = 1 4 8 27 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpdaWcaaqaamaalaaabaGaaiikaiaaigdacaGGPaWaaWbaaSqabeaa caaIYaaaaaGcbaGaaiikaiaaikdacaGGPaWaaWbaaSqabeaacaaIYa aaaaaaaOqaamaalaaabaGaaiikaiaaikdacaGGPaWaaWbaaSqabeaa caaIZaaaaaGcbaGaaiikaiaaiodacaGGPaWaaWbaaSqabeaacaaIZa aaaaaaaaaakeaacqGH9aqpdaWcaaqaamaalaaabaGaaGymaaqaaiaa isdaaaaabaWaaSaaaeaacaaI4aaabaGaaGOmaiaaiEdaaaaaaaaaaa@4825@

= 1 4 × 27 8 = 27 4×8 = 27 32 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpdaWcaaqaaiaaigdaaeaacaaI0aaaaiabgEna0oaalaaabaGaaGOm aiaaiEdaaeaacaaI4aaaaaqaaiabg2da9maalaaabaGaaGOmaiaaiE daaeaacaaI0aGaey41aqRaaGioaaaaaeaacqGH9aqpdaWcaaqaaiaa ikdacaaI3aaabaGaaG4maiaaikdaaaaaaaa@470A@

The reciprocal is 32 27 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaGaaGOmaaqaaiaaikdacaaI3aaaaaaa@38DE@

Question: 72

Find the value of:

a.    7 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4namaaCa aaleqabaGaaGimaaaaaaa@3781@

b.   7 7 ÷ 7 7 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4namaaCa aaleqabaGaaG4naaaakiabgEpa4kaaiEdadaahaaWcbeqaaiaaiEda aaaaaa@3B7C@

c.    (7) 2×768 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabgk HiTiaaiEdacaGGPaWaaWbaaSqabeaacaaIYaGaey41aqRaaG4naiab gkHiTiaaiAdacqGHsislcaaI4aaaaaaa@3FFC@

d.   ( 2 0 + 3 0 + 4 0 ) ( 4 0 3 0 2 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaik dadaahaaWcbeqaaiaaicdaaaGccqGHRaWkcaaIZaWaaWbaaSqabeaa caaIWaaaaOGaey4kaSIaaGinamaaCaaaleqabaGaaGimaaaakiaacM cacaqGGaGaaiikaiaaisdadaahaaWcbeqaaiaaicdaaaGccqGHsisl caaIZaWaaWbaaSqabeaacaaIWaaaaOGaeyOeI0IaaGOmamaaCaaale qabaGaaGimaaaakiaacMcaaaa@46DF@

e.    2×3×4÷ 2 0 × 3 0 × 4 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiabgE na0kaaiodacqGHxdaTcaaI0aGaey49aGRaaGOmamaaCaaaleqabaGa aGimaaaakiabgEna0kaaiodadaahaaWcbeqaaiaaicdaaaGccqGHxd aTcaaI0aWaaWbaaSqabeaacaaIWaaaaaaa@47A6@

f. ( 8 0 2 0 )×( 8 0 + 2 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaiI dadaahaaWcbeqaaiaaicdaaaGccqGHsislcaaIYaWaaWbaaSqabeaa caaIWaaaaOGaaiykaiabgEna0kaacIcacaaI4aWaaWbaaSqabeaaca aIWaaaaOGaey4kaSIaaGOmamaaCaaaleqabaGaaGimaaaakiaacMca aaa@4330@

Solution

a.    7 0 =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4namaaCa aaleqabaGaaGimaaaakiabg2da9iaaigdaaaa@394B@ , as we know that any number to the power of zero is one.

b.   7 7  ÷ 7 7 = 7 7 7 7 = 7 77 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaaI3a WaaWbaaSqabeaacaaI3aGaaiiOaaaakiabgEpa4kaaiEdadaahaaWc beqaaiaaiEdaaaaakeaacqGH9aqpdaWcaaqaaiaaiEdadaahaaWcbe qaaiaaiEdaaaaakeaacaaI3aWaaWbaaSqabeaacaaI3aaaaaaaaOqa aiabg2da9iaaiEdadaahaaWcbeqaaiaaiEdacqGHsislcaaI3aaaaa aaaa@459B@

= 7 0 =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaaI3aWaaWbaaSqabeaacaaIWaaaaaGcbaGaeyypa0JaaGymaaaa aa@3A57@

c.    ( 7 ) 2×768 = (7) 1414 = ( 7 ) 0 =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaqada qaaiabgkHiTiaaiEdaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikda cqGHxdaTcaaI3aGaeyOeI0IaaGOnaiabgkHiTiaaiIdaaaaakeaacq GH9aqpcaGGOaGaeyOeI0IaaG4naiaacMcadaahaaWcbeqaaiaaigda caaI0aGaeyOeI0IaaGymaiaaisdaaaaakeaacqGH9aqpdaqadaqaai abgkHiTiaaiEdaaiaawIcacaGLPaaadaahaaWcbeqaaiaaicdaaaaa keaacqGH9aqpcaaIXaaaaaa@4F50@

d.   ( 2 0 + 3 0 + 4 0 )( 4 0 3 0 2 0 ) =(1+1+1)(111) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaGGOa GaaGOmamaaCaaaleqabaGaaGimaaaakiabgUcaRiaaiodadaahaaWc beqaaiaaicdaaaGccqGHRaWkcaaI0aWaaWbaaSqabeaacaaIWaaaaO GaaiykaiaacIcacaaI0aWaaWbaaSqabeaacaaIWaaaaOGaeyOeI0Ia aG4mamaaCaaaleqabaGaaGimaaaakiabgkHiTiaaikdadaahaaWcbe qaaiaaicdaaaGccaGGPaaabaGaeyypa0JaaiikaiaaigdacqGHRaWk caaIXaGaey4kaSIaaGymaiaacMcacaGGOaGaaGymaiabgkHiTiaaig dacqGHsislcaaIXaGaaiykaaaaaa@51FA@

=( 3 )(1) =3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpdaqadaqaaiaaiodaaiaawIcacaGLPaaacaGGOaGaeyOeI0IaaGym aiaacMcaaeaacqGH9aqpcqGHsislcaaIZaaaaaa@3EDC@

e.    2×3×4÷ 2 0 × 3 0 × 4 0 =2×3×4÷1×1×1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaaIYa Gaey41aqRaaG4maiabgEna0kaaisdacqGH3daUcaaIYaWaaWbaaSqa beaacaaIWaaaaOGaey41aqRaaG4mamaaCaaaleqabaGaaGimaaaaki abgEna0kaaisdadaahaaWcbeqaaiaaicdaaaaakeaacqGH9aqpcaaI YaGaey41aqRaaG4maiabgEna0kaaisdacqGH3daUcaaIXaGaey41aq RaaGymaiabgEna0kaaigdaaaaa@57BB@

= 2×3×4 1×1×1 =2×3×4 =24 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpdaWcaaqaaiaaikdacqGHxdaTcaaIZaGaey41aqRaaGinaaqaaiaa igdacqGHxdaTcaaIXaGaey41aqRaaGymaaaaaeaacqGH9aqpcaaIYa Gaey41aqRaaG4maiabgEna0kaaisdaaeaacqGH9aqpcaaIYaGaaGin aaaaaa@4DA4@

f.     ( 8 0 2 0 )×( 8 0 + 2 0 ) =(11)×( 1+1 ) =0×2=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaGGOa GaaGioamaaCaaaleqabaGaaGimaaaakiabgkHiTiaaikdadaahaaWc beqaaiaaicdaaaGccaGGPaGaey41aqRaaiikaiaaiIdadaahaaWcbe qaaiaaicdaaaGccqGHRaWkcaaIYaWaaWbaaSqabeaacaaIWaaaaOGa aiykaaqaaiabg2da9iaacIcacaaIXaGaeyOeI0IaaGymaiaacMcacq GHxdaTdaqadaqaaiaaigdacqGHRaWkcaaIXaaacaGLOaGaayzkaaaa baGaeyypa0JaaGimaiabgEna0kaaikdacqGH9aqpcaaIWaaaaaa@5444@

Question: 73

Find the value of n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36CC@ , where n is an integer and

2 n5 × 6 2n4 = 1 12 4 ×2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaamOBaiabgkHiTiaaiwdaaaGccqGHxdaTcaaI2aWaaWba aSqabeaacaaIYaGaamOBaiabgkHiTiaaisdaaaGccaaMc8Uaaeypam aalaaabaGaaGymaaqaaiaaigdacaaIYaWaaWbaaSqabeaacaaI0aaa aOGaey41aqRaaGOmaaaacaqGUaGaaeiiaaaa@497B@

Solution

Considering the given equation,    

2 n5 × 6 2 n4 = 1 12 4 ×2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaamOBaiabgkHiTiaaiwdaaaGccqGHxdaTcaaI2aWaaWba aSqabeaacaaIYaaaaOWaaWbaaSqabeaacaWGUbGaeyOeI0IaaGinaa aakiabg2da9maalaaabaGaaGymaaqaaiaaigdacaaIYaWaaWbaaSqa beaacaaI0aaaaOGaey41aqRaaGOmaaaaaaa@4719@

2 n 2 5 × 6 2n 6 4 = 1 12 4 ×2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIYaWaaWbaaSqabeaacaWGUbaaaaGcbaGaaGOmamaaCaaaleqabaGa aGynaaaaaaGccqGHxdaTdaWcaaqaaiaaiAdadaahaaWcbeqaaiaaik dacaWGUbaaaaGcbaGaaGOnamaaCaaaleqabaGaaGinaaaaaaGccqGH 9aqpdaWcaaqaaiaaigdaaeaacaaIXaGaaGOmamaaCaaaleqabaGaaG inaaaakiabgEna0kaaikdaaaaaaa@4712@

2 n × 6 2n 2 5 × 6 4 = 1 (2×6) 4 ×2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIYaWaaWbaaSqabeaacaWGUbaaaOGaey41aqRaaGOnamaaCaaaleqa baGaaGOmaiaad6gaaaaakeaacaaIYaWaaWbaaSqabeaacaaI1aaaaO Gaey41aqRaaGOnamaaCaaaleqabaGaaGinaaaaaaGccqGH9aqpdaWc aaqaaiaaigdaaeaacaGGOaGaaGOmaiabgEna0kaaiAdacaGGPaWaaW baaSqabeaacaaI0aaaaOGaey41aqRaaGOmaaaaaaa@4C8E@

2 n × ( 6 2 ) n = ( 2 5 × 6 4 ) 2 4 × 6 4 ×2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaamOBaaaakiabgEna0kaacIcacaaI2aWaaWbaaSqabeaa caaIYaaaaOGaaiykamaaCaaaleqabaGaamOBaaaakiabg2da9maala aabaGaaiikaiaaikdadaahaaWcbeqaaiaaiwdaaaGccqGHxdaTcaaI 2aWaaWbaaSqabeaacaaI0aaaaOGaaiykaaqaaiaaikdadaahaaWcbe qaaiaaisdaaaGccqGHxdaTcaaI2aWaaWbaaSqabeaacaaI0aaaaOGa ey41aqRaaGOmaaaaaaa@4E48@

2 n × 36 n = 2 5 × 6 4 2 5 × 6 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaamOBaaaakiabgEna0kaaiodacaaI2aWaaWbaaSqabeaa caWGUbaaaOGaeyypa0ZaaSaaaeaacaaIYaWaaWbaaSqabeaacaaI1a aaaOGaey41aqRaaGOnamaaCaaaleqabaGaaGinaaaaaOqaaiaaikda daahaaWcbeqaaiaaiwdaaaGccqGHxdaTcaaI2aWaaWbaaSqabeaaca aI0aaaaaaaaaa@4884@

2 n × 36 n =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaamOBaaaakiabgEna0kaaiodacaaI2aWaaWbaaSqabeaa caWGUbaaaOGaeyypa0JaaGymaaaa@3E3E@

( 2×36 ) n =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIYaGaey41aqRaaG4maiaaiAdaaiaawIcacaGLPaaadaahaaWcbeqa aiaad6gaaaGccqGH9aqpcaaIXaaaaa@3E9C@

72 n = 72 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4naiaaik dadaahaaWcbeqaaiaad6gaaaGccqGH9aqpcaaI3aGaaGOmamaaCaaa leqabaGaaGimaaaaaaa@3BE9@

n=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg2 da9iaaicdaaaa@388C@

Question: 74

Express the following in usual form:

a.    8.01× 10 7 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaac6 cacaaIWaGaaGymaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaI 3aaaaaaa@3D3C@

b.   1.75× 10 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaac6 cacaaI3aGaaGynaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacqGH sislcaaIZaaaaaaa@3E29@

Solution

a.    The given number 8.01× 10 7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaac6 cacaaIWaGaaGymaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaI 3aaaaaaa@3D3B@  can be written as

= 801 100 ×100000000 =80100000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpdaWcaaqaaiaaiIdacaaIWaGaaGymaaqaaiaaigdacaaIWaGaaGim aaaacqGHxdaTcaaIXaGaaGimaiaaicdacaaIWaGaaGimaiaaicdaca aIWaGaaGimaiaaicdaaeaacqGH9aqpcaaI4aGaaGimaiaaigdacaaI WaGaaGimaiaaicdacaaIWaGaaGimaaaaaa@4ADB@

b.   The given number 1.75× 10 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaac6 cacaaI3aGaaGynaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacqGH sislcaaIZaaaaaaa@3E28@  can be written as

= 175 100 × 1 10 3 = 175 100000 =0.00175 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpdaWcaaqaaiaaigdacaaI3aGaaGynaaqaaiaaigdacaaIWaGaaGim aaaacqGHxdaTdaWcaaqaaiaaigdaaeaacaaIXaGaaGimamaaCaaale qabaGaaG4maaaaaaaakeaacqGH9aqpdaWcaaqaaiaaigdacaaI3aGa aGynaaqaaiaaigdacaaIWaGaaGimaiaaicdacaaIWaGaaGimaaaaae aacqGH9aqpcaaIWaGaaiOlaiaaicdacaaIWaGaaGymaiaaiEdacaaI 1aaaaaa@4E7A@

Question: 75

Find the value of

a.    2 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaGynaaaaaaa@3781@

b.   ( 3 5 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq GHsislcaaIZaWaaWbaaSqabeaacaaI1aaaaaGccaGLOaGaayzkaaaa aa@3A02@

c.    ( 4 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Yaae WaaeaacqGHsislcaaI0aaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI 0aaaaaaa@3AE5@

Solution

We have studied that, a n =a×a×a××a     ( n times ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaamOBaaaakiabg2da9iaadggacqGHxdaTcaWGHbGaey41 aqRaamyyaiabgEna0kabgAci8kabgAci8kabgEna0kaadggacaGGGc GaaiiOaiaacckacaGGGcGaaiiOamaabmaabaGaamOBaiaabccacaqG 0bGaaeyAaiaab2gacaqGLbGaae4CaaGaayjkaiaawMcaaaaa@5582@

a.      2 5 =2×2×2×2×2 =32 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaaIYa WaaWbaaSqabeaacaaI1aaaaaGcbaGaeyypa0JaaGOmaiabgEna0kaa ikdacqGHxdaTcaaIYaGaey41aqRaaGOmaiabgEna0kaaikdaaeaacq GH9aqpcaaIZaGaaGOmaaaaaa@471E@

b.   (3) 5 = (1) 5 × 3 5 =1×3×3×3×3×3 =243 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaGGOa GaeyOeI0IaaG4maiaacMcadaahaaWcbeqaaiaaiwdaaaaakeaacqGH 9aqpcaGGOaGaeyOeI0IaaGymaiaacMcadaahaaWcbeqaaiaaiwdaaa GccqGHxdaTcaaIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaeyypa0Ja eyOeI0IaaGymaiabgEna0kaaiodacqGHxdaTcaaIZaGaey41aqRaaG 4maiabgEna0kaaiodacqGHxdaTcaaIZaaabaGaeyypa0JaeyOeI0Ia aGOmaiaaisdacaaIZaaaaaa@579C@

c.    (4) 4 =[(4)×(4)×(4)×(4)] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGHsi slcaGGOaGaeyOeI0IaaGinaiaacMcadaahaaWcbeqaaiaaisdaaaaa keaacqGH9aqpcqGHsislcaGGBbGaaiikaiabgkHiTiaaisdacaGGPa Gaey41aqRaaiikaiabgkHiTiaaisdacaGGPaGaey41aqRaaiikaiab gkHiTiaaisdacaGGPaGaey41aqRaaiikaiabgkHiTiaaisdacaGGPa Gaaiyxaaaaaa@50CC@

=[ (1) 4 (4×4×4×4)] =( 256 ) =256 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcqGHsislcaGGBbGaaiikaiabgkHiTiaaigdacaGGPaWaaWbaaSqa beaacaaI0aaaaOGaaiikaiaaisdacqGHxdaTcaaI0aGaey41aqRaaG inaiabgEna0kaaisdacaGGPaGaaiyxaaqaaiabg2da9iabgkHiTmaa bmaabaGaaGOmaiaaiwdacaaI2aaacaGLOaGaayzkaaaabaGaeyypa0 JaeyOeI0IaaGOmaiaaiwdacaaI2aaaaaa@5203@

Question: 76

Express the following in exponential form:

a.    3×3×3×a×a×a×a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabgE na0kaaiodacqGHxdaTcaaIZaGaey41aqRaamyyaiabgEna0kaadgga cqGHxdaTcaWGHbGaey41aqRaamyyaaaa@4831@

b.   a×a×b×b×b×c×c×c×c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgE na0kaadggacqGHxdaTcaWGIbGaey41aqRaamOyaiabgEna0kaadkga cqGHxdaTcaWGJbGaey41aqRaam4yaiabgEna0kaadogacqGHxdaTca WGJbaaaa@4EB1@

c.    s×s×t×t×s×s×t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiabgE na0kaadohacqGHxdaTcaWG0bGaey41aqRaamiDaiabgEna0kaadoha cqGHxdaTcaWGZbGaey41aqRaamiDaaaa@492D@

Solution

We have studied that, a n =a×a×a×......×a (n times)= a n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaamOBaaaakiabg2da9iaadggacqGHxdaTcaWGHbGaey41 aqRaamyyaiabgEna0kaac6cacaGGUaGaaiOlaiaac6cacaGGUaGaai OlaiabgEna0kaadggacaqGGaGaaiikaiaad6gacaqGGaGaaeiDaiaa bMgacaqGTbGaaeyzaiaabohacaGGPaGaeyypa0JaamyyamaaCaaale qabaGaamOBaaaaaaa@545D@

a.    3×3×3×a×a×a×a = 3 3 · a 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaaIZa Gaey41aqRaaG4maiabgEna0kaaiodacqGHxdaTcaWGHbGaey41aqRa amyyaiabgEna0kaadggacqGHxdaTcaWGHbaabaGaeyypa0JaaG4mam aaCaaaleqabaGaaG4maaaakiabl+y6NjaaysW7caWGHbWaaWbaaSqa beaacaaI0aaaaaaaaa@50BC@

b.   a×a×b×b×b×c×c×c×c = a 2 · b 3 · c 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaWGHb Gaey41aqRaamyyaiabgEna0kaadkgacqGHxdaTcaWGIbGaey41aqRa amOyaiabgEna0kaadogacqGHxdaTcaWGJbGaey41aqRaam4yaiabgE na0kaadogaaeaacqGH9aqpcaWGHbWaaWbaaSqabeaacaaIYaaaaOGa eS4JPFMaaGjbVlaadkgadaahaaWcbeqaaiaaiodaaaGccqWIpM+zca aMe8Uaam4yamaaCaaaleqabaGaaGinaaaaaaaa@5D3E@

c.    s×s×t×t×s×s×t =s×s×s×s×t×t×t = s 4 · t 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaWGZb Gaey41aqRaam4CaiabgEna0kaadshacqGHxdaTcaWG0bGaey41aqRa am4CaiabgEna0kaadohacqGHxdaTcaWG0baabaGaeyypa0Jaam4Cai abgEna0kaadohacqGHxdaTcaWGZbGaey41aqRaam4CaiabgEna0kaa dshacqGHxdaTcaWG0bGaey41aqRaamiDaaqaaiabg2da9iaadohada ahaaWcbeqaaiaaisdaaaGccqWIpM+zcaaMe8UaamiDamaaCaaaleqa baGaaG4maaaaaaaa@6662@

Question: 77

How many times of 30 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaaic daaaa@3750@  must be added together to get a sum equal to 30 7 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaaic dadaahaaWcbeqaaiaaiEdaaaaaaa@383E@ ?

Solution

Let n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36CC@  be the number of times that 30 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaaic daaaa@3750@  must be added together to get a sum equal to 30 7 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaaic dadaahaaWcbeqaaiaaiEdaaaGccaGGUaaaaa@38FA@

Therefore, we can write the above statement as -  

30+30++30= 30 7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaaic dacqGHRaWkcaaIZaGaaGimaiabgUcaRiabgAci8kabgUcaRiaaioda caaIWaGaeyypa0JaaG4maiaaicdadaahaaWcbeqaaiaaiEdaaaaaaa@41DC@          ( n times ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGUbGaaeiiaiaabshacaqGPbGaaeyBaiaabwgacaqGZbaacaGLOaGa ayzkaaaaaa@3DA8@

30×n= 30 7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaaic dacqGHxdaTcaWGUbGaeyypa0JaaG4maiaaicdadaahaaWcbeqaaiaa iEdaaaaaaa@3DC4@

30×n 30 = 30 7 30 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaGaaGimaiabgEna0kaad6gaaeaacaaIZaGaaGimaaaacqGH9aqp daWcaaqaaiaaiodacaaIWaWaaWbaaSqabeaacaaI3aaaaaGcbaGaaG 4maiaaicdaaaaaaa@40DC@

n= 30 71 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg2 da9iaaiodacaaIWaWaaWbaaSqabeaacaaI3aGaeyOeI0IaaGymaaaa aaa@3BDE@

n= 30 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg2 da9iaaiodacaaIWaWaaWbaaSqabeaacaaI2aaaaaaa@3A35@

Hence, if 30 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaaic daaaa@374F@  is added 30 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaaic dadaahaaWcbeqaaiaaiAdaaaaaaa@383C@  times, then we get 30 7 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaaic dadaahaaWcbeqaaiaaiEdaaaGccaGGUaaaaa@38F9@

Question: 78

Express each of the following numbers using exponential notations:

a.    1024 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic dacaaIYaGaaGinaaaa@38C8@

b.   1029 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic dacaaIYaGaaGyoaaaa@38CD@

c.    144 875 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaGaaGinaiaaisdaaeaacaaI4aGaaG4naiaaiwdaaaaaaa@3A62@

Solution

a.    The number 1024 using prime factorisation can be written as -
1024=2×2×2×2×2×2×2×2×2×2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic dacaaIYaGaaGinaiabg2da9iaaikdacqGHxdaTcaaIYaGaey41aqRa aGOmaiabgEna0kaaikdacqGHxdaTcaaIYaGaey41aqRaaGOmaiabgE na0kaaikdacqGHxdaTcaaIYaGaey41aqRaaGOmaiabgEna0kaaikda aaa@53F4@
  = 2 10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG OmamaaCaaaleqabaGaaGymaiaaicdaaaaaaa@393C@

b.   The number 1029 using prime factorisation can be written as -
=3×7×7×7 =3× 7 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaaIZaGaey41aqRaaG4naiabgEna0kaaiEdacqGHxdaTcaaI3aaa baGaeyypa0JaaG4maiabgEna0kaaiEdadaahaaWcbeqaaiaaiodaaa aaaaa@45AE@

c.    The number  144 875 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaGaaGinaiaaisdaaeaacaaI4aGaaG4naiaaiwdaaaaaaa@3A61@  using prime factorisation can be written as -
 
= 2×2×2×2×3×3 5×5×5×7 = 2 4 × 3 2 5 3 × 7 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpdaWcaaqaaiaaikdacqGHxdaTcaaIYaGaey41aqRaaGOmaiabgEna 0kaaikdacqGHxdaTcaaIZaGaey41aqRaaG4maaqaaiaaiwdacqGHxd aTcaaI1aGaey41aqRaaGynaiabgEna0kaaiEdaaaaabaGaeyypa0Za aSaaaeaacaaIYaWaaWbaaSqabeaacaaI0aaaaOGaey41aqRaaG4mam aaCaaaleqabaGaaGOmaaaaaOqaaiaaiwdadaahaaWcbeqaaiaaioda aaGccqGHxdaTcaaI3aWaaWbaaSqabeaacaaIXaaaaaaaaaaa@5B15@

Question: 79

Identify the greater number, in each of the following:

a.    2 6  or  6 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaGOnaaaakiaabccacaqGVbGaaeOCaiaaykW7caqGGaGa aGOnamaaCaaaleqabaGaaGOmaaaaaaa@3DEC@

b.   2 9 or  9 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaGyoaaaakiaaykW7caaMc8Uaae4BaiaabkhacaqGGaGa aGPaVlaaiMdadaahaaWcbeqaaiaaikdaaaaaaa@4066@

c.    7.9× 10 4 or5.28× 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4naiaac6 cacaaI5aGaey41aqRaaGymaiaaicdadaahaaWcbeqaaiaaisdaaaGc caaMc8Uaae4BaiaabkhacaaMc8UaaGPaVlaaiwdacaGGUaGaaGOmai aaiIdacqGHxdaTcaaIXaGaaGimamaaCaaaleqabaGaaGynaaaaaaa@4A7F@

Solution

a.    We know that, 2 6 =2×2×2×2×2×2=64 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaGOnaaaakiabg2da9iaaikdacqGHxdaTcaaIYaGaey41 aqRaaGOmaiabgEna0kaaikdacqGHxdaTcaaIYaGaey41aqRaaGOmai abg2da9iaaiAdacaaI0aaaaa@49F0@  and 6 2 =6×6=36 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnamaaCa aaleqabaGaaGOmaaaakiabg2da9iaaiAdacqGHxdaTcaaI2aGaeyyp a0JaaG4maiaaiAdaaaa@3EAB@  
S
o, 2 6 > 6 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOmamaaCa aaleqabaGaaeOnaaaakiabg6da+iaabAdadaahaaWcbeqaaiaabAda aaaaaa@3A24@

b.   We know that, 2 9 =2×2×2×2×2×2×2×2×2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaGyoaaaakiabg2da9iaaikdacqGHxdaTcaaIYaGaey41 aqRaaGOmaiabgEna0kaaikdacqGHxdaTcaaIYaGaey41aqRaaGOmai abgEna0kaaikdacqGHxdaTcaaIYaGaey41aqRaaGOmaaaa@4FE8@

=512 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ynaiaaigdacaaIYaaaaa@3914@  and 9 2 =9×9=81 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoamaaCa aaleqabaGaaGOmaaaakiabg2da9iaaiMdacqGHxdaTcaaI5aGaeyyp a0JaaGioaiaaigdaaaa@3EB4@

So, 2 9 >  9 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaGyoaaaakiaaykW7caaMc8UaeyOpa4JaaeiiaiaaykW7 caaI5aWaaWbaaSqabeaacaaIYaaaaaaa@3F87@

c.    We know that, 7.9× 10 4 =7.9×10000=79000 and MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4naiaac6 cacaaI5aGaey41aqRaaGymaiaaicdadaahaaWcbeqaaiaaisdaaaGc cqGH9aqpcaaI3aGaaiOlaiaaiMdacqGHxdaTcaaIXaGaaGimaiaaic dacaaIWaGaaGimaiabg2da9iaaiEdacaaI5aGaaGimaiaaicdacaaI WaGaaeiiaiaabggacaqGUbGaaeizaaaa@4D9C@

5.28=5.28×100000=528000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaiaac6 cacaaIYaGaaGioaiabg2da9iaaiwdacaGGUaGaaGOmaiaaiIdacqGH xdaTcaaIXaGaaGimaiaaicdacaaIWaGaaGimaiaaicdacqGH9aqpca aI1aGaaGOmaiaaiIdacaaIWaGaaGimaiaaicdaaaa@48A1@

So, 5.28× 10 5 >7.9× 10 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaiaac6 cacaaIYaGaaGioaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaI 1aaaaOGaeyOpa4JaaG4naiaac6cacaaI5aGaey41aqRaaGymaiaaic dadaahaaWcbeqaaiaaisdaaaaaaa@44FF@

Question: 80

Express each of the following as a product of powers of their prime factors:

a.    9000 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoaiaaic dacaaIWaGaaGimaaaa@38CA@

b.   2025 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaaic dacaaIYaGaaGynaaaa@38CA@

c.    800 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaaic dacaaIWaaaaa@380F@

Solution

The number 9000 using prime factorisation can be written as -

9000= 2 3 × 3 2 × 5 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoaiaaic dacaaIWaGaaGimaiabg2da9iaaikdadaahaaWcbeqaaiaaiodaaaGc cqGHxdaTcaaIZaWaaWbaaSqabeaacaaIYaaaaOGaey41aqRaaGynam aaCaaaleqabaGaaG4maaaaaaa@4307@

The number 2025 using prime factorisation can be written as -

The number 800 using prime factorisation can be written as -

Question: 81

Express each of the following in single exponential form:

a.    2 3 × 3 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaG4maaaakiabgEna0kaaiodadaahaaWcbeqaaiaaioda aaaaaa@3B46@

b.   2 4 × 4 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaGinaaaakiabgEna0kaaisdadaahaaWcbeqaaiaaikda aaaaaa@3B47@

c.    5 2 × 7 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynamaaCa aaleqabaGaaGOmaaaakiabgEna0kaaiEdadaahaaWcbeqaaiaaikda aaaaaa@3B4B@

d.   (5) 5 ×(5) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabgk HiTiaaiwdacaGGPaWaaWbaaSqabeaacaaI1aaaaOGaey41aqRaaiik aiabgkHiTiaaiwdacaGGPaaaaa@3EEF@

e.    (3) 3 × (10) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabgk HiTiaaiodacaGGPaWaaWbaaSqabeaacaaIZaaaaOGaey41aqRaaiik aiabgkHiTiaaigdacaaIWaGaaiykamaaCaaaleqabaGaaG4maaaaaa a@408B@

f. ( 11 ) 2 × ( 2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq GHsislcaaIXaGaaGymaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOm aaaakiabgEna0oaabmaabaGaeyOeI0IaaGOmaaGaayjkaiaawMcaam aaCaaaleqabaGaaGOmaaaaaaa@40EA@

Solution

a.    We have,
2 3 × 3 3 = (2×3) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaaIYa WaaWbaaSqabeaacaaIZaaaaOGaey41aqRaaG4mamaaCaaaleqabaGa aG4maaaaaOqaaiabg2da9iaacIcacaaIYaGaey41aqRaaG4maiaacM cadaahaaWcbeqaaiaaiodaaaaaaaa@422F@

b.   We have,
2 4 × 4 2 = 2 4 × ( 2 2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaaIYa WaaWbaaSqabeaacaaI0aaaaOGaey41aqRaaGinamaaCaaaleqabaGa aGOmaaaaaOqaaiabg2da9iaaikdadaahaaWcbeqaaiaaisdaaaGccq GHxdaTcaGGOaGaaGOmamaaCaaaleqabaGaaGOmaaaakiaacMcadaah aaWcbeqaaiaaikdaaaaaaaa@4416@

c.    We have,

5 2 × 7 2 = (5×7) 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaaI1a WaaWbaaSqabeaacaaIYaaaaOGaey41aqRaaG4namaaCaaaleqabaGa aGOmaaaaaOqaaiabg2da9iaacIcacaaI1aGaey41aqRaaG4naiaacM cadaahaaWcbeqaaiaaikdaaaGccaGGUaaaaaa@42F6@  

= 35 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG 4maiaaiwdadaahaaWcbeqaaiaaikdaaaaaaa@3943@

d.   We have,

(5) 5 ×(5) = (5) 5+1 = (5) 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaGGOa GaeyOeI0IaaGynaiaacMcadaahaaWcbeqaaiaaiwdaaaGccqGHxdaT caGGOaGaeyOeI0IaaGynaiaacMcaaeaacqGH9aqpcaGGOaGaeyOeI0 IaaGynaiaacMcadaahaaWcbeqaaiaaiwdacqGHRaWkcaaIXaaaaaGc baGaeyypa0JaaiikaiabgkHiTiaaiwdacaGGPaWaaWbaaSqabeaaca aI2aaaaaaaaa@4A8C@

= (1×5) 6 = (1) 6 × (5) 6 =1× 5 6 = 5 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaGGOaGaeyOeI0IaaGymaiabgEna0kaaiwdacaGGPaWaaWbaaSqa beaacaaI2aaaaaGcbaGaeyypa0JaaiikaiabgkHiTiaaigdacaGGPa WaaWbaaSqabeaacaaI2aaaaOGaey41aqRaaiikaiaaiwdacaGGPaWa aWbaaSqabeaacaaI2aaaaaGcbaGaeyypa0JaaGymaiabgEna0kaaiw dadaahaaWcbeqaaiaaiAdaaaaakeaacqGH9aqpcaaI1aWaaWbaaSqa beaacaaI2aaaaaaaaa@5018@

e.    We have,
(3) 3 × (10) 3 = [(3)×(10)] 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaGGOa GaeyOeI0IaaG4maiaacMcadaahaaWcbeqaaiaaiodaaaGccqGHxdaT caGGOaGaeyOeI0IaaGymaiaaicdacaGGPaWaaWbaaSqabeaacaaIZa aaaaGcbaGaeyypa0Jaai4waiaacIcacqGHsislcaaIZaGaaiykaiab gEna0kaacIcacqGHsislcaaIXaGaaGimaiaacMcacaGGDbWaaWbaaS qabeaacaaIZaaaaaaaaa@4D20@

= (30) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaai ikaiaaiodacaaIWaGaaiykamaaCaaaleqabaGaaG4maaaaaaa@3A98@

f.     We have,
(11) 2 × (2) 2 = [(11)×(2)] 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaGGOa GaeyOeI0IaaGymaiaaigdacaGGPaWaaWbaaSqabeaacaaIYaaaaOGa ey41aqRaaiikaiabgkHiTiaaikdacaGGPaWaaWbaaSqabeaacaaIYa aaaaGcbaGaeyypa0Jaai4waiaacIcacqGHsislcaaIXaGaaGymaiaa cMcacqGHxdaTcaGGOaGaeyOeI0IaaGOmaiaacMcacaGGDbWaaWbaaS qabeaacaaIYaaaaaaaaa@4D1D@

Question: 82

Express the following numbers in standard form:

a.    76,47,000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4naiaaiA dacaGGSaGaaGinaiaaiEdacaGGSaGaaGimaiaaicdacaaIWaaaaa@3C66@

b.   8,19,00,000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaacY cacaaIXaGaaGyoaiaacYcacaaIWaGaaGimaiaacYcacaaIWaGaaGim aiaaicdaaaa@3DCA@

c.    5,83,00,00,00,000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaiaacY cacaaI4aGaaG4maiaacYcacaaIWaGaaGimaiaacYcacaaIWaGaaGim aiaacYcacaaIWaGaaGimaiaacYcacaaIWaGaaGimaiaaicdaaaa@4210@

d.   24 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaais daaaa@3752@  billion

Solution

a.    We have, 76,47,000=7647000.00 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4naiaaiA dacaGGSaGaaGinaiaaiEdacaGGSaGaaGimaiaaicdacaaIWaGaeyyp a0JaaG4naiaaiAdacaaI0aGaaG4naiaaicdacaaIWaGaaGimaiaac6 cacaaIWaGaaGimaaaa@44C0@

We have studied that, a number in standard form is written as a ×  10 k , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey41aqRaae iiaiaaigdacaaIWaWaaWbaaSqabeaacaWGRbaaaOGaaiilaaaa@3BDE@

where a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36BF@  is the terminating decimal such that 1a<10 and k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgs MiJkaadggacqGH8aapcaaIXaGaaGimaiaabccacaqGHbGaaeOBaiaa bsgacaqGGaGaam4Aaaaa@4099@  is any integer.

So, 7647000=7647× 10 3 =7.647× 10 3 × 10 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4naiaaiA dacaaI0aGaaG4naiaaicdacaaIWaGaaGimaiabg2da9iaaiEdacaaI 2aGaaGinaiaaiEdacqGHxdaTcaaIXaGaaGimamaaCaaaleqabaGaaG 4maaaakiabg2da9iaaiEdacaGGUaGaaGOnaiaaisdacaaI3aGaey41 aqRaaGymaiaaicdadaahaaWcbeqaaiaaiodaaaGccqGHxdaTcaaIXa GaaGimamaaCaaaleqabaGaaG4maaaaaaa@513A@

=7.647× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG 4naiaac6cacaaI2aGaaGinaiaaiEdacqGHxdaTcaaIXaGaaGimamaa CaaaleqabaGaaGOnaaaaaaa@3F09@

Similarly,

b.   8,19,00,000=81900000.00=819× 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaacY cacaaIXaGaaGyoaiaacYcacaaIWaGaaGimaiaacYcacaaIWaGaaGim aiaaicdacqGH9aqpcaaI4aGaaGymaiaaiMdacaaIWaGaaGimaiaaic dacaaIWaGaaGimaiaac6cacaaIWaGaaGimaiabg2da9iaaiIdacaaI XaGaaGyoaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaI1aaaaa aa@4E96@

=8.19× 10 2 × 10 5 =8.19× 10 7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ioaiaac6cacaaIXaGaaGyoaiabgEna0kaaigdacaaIWaWaaWbaaSqa beaacaaIYaaaaOGaey41aqRaaGymaiaaicdadaahaaWcbeqaaiaaiw daaaGccqGH9aqpcaaI4aGaaiOlaiaaigdacaaI5aGaey41aqRaaGym aiaaicdadaahaaWcbeqaaiaaiEdaaaaaaa@4B43@

c.    5,83,00,00,00,000=583000000000.00 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaiaacY cacaaI4aGaaG4maiaacYcacaaIWaGaaGimaiaacYcacaaIWaGaaGim aiaacYcacaaIWaGaaGimaiaacYcacaaIWaGaaGimaiaaicdacqGH9a qpcaaI1aGaaGioaiaaiodacaaIWaGaaGimaiaaicdacaaIWaGaaGim aiaaicdacaaIWaGaaGimaiaaicdacaGGUaGaaGimaiaaicdaaaa@4E04@

=583× 10 9 =5.83× 10 2 × 10 9 =5.83× 10 11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ynaiaaiIdacaaIZaGaey41aqRaaGymaiaaicdadaahaaWcbeqaaiaa iMdaaaGccqGH9aqpcaaI1aGaaiOlaiaaiIdacaaIZaGaey41aqRaaG ymaiaaicdadaahaaWcbeqaaiaaikdaaaGccqGHxdaTcaaIXaGaaGim amaaCaaaleqabaGaaGyoaaaakiabg2da9iaaiwdacaGGUaGaaGioai aaiodacqGHxdaTcaaIXaGaaGimamaaCaaaleqabaGaaGymaiaaigda aaaaaa@53C2@

d.   24 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaais daaaa@3752@   billion MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOyaiaabM gacaqGSbGaaeiBaiaabMgacaqGVbGaaeOBaaaa@3C56@   =24,00,00,00,000=24× 10 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG OmaiaaisdacaGGSaGaaGimaiaaicdacaGGSaGaaGimaiaaicdacaGG SaGaaGimaiaaicdacaGGSaGaaGimaiaaicdacaaIWaGaeyypa0JaaG OmaiaaisdacqGHxdaTcaaIXaGaaGimamaaCaaaleqabaGaaGyoaaaa aaa@489E@

=2.4× 10 1 × 10 9 =2.4× 10 10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG Omaiaac6cacaaI0aGaey41aqRaaGymaiaaicdadaahaaWcbeqaaiaa igdaaaGccqGHxdaTcaaIXaGaaGimamaaCaaaleqabaGaaGyoaaaaki abg2da9iaaikdacaGGUaGaaGinaiabgEna0kaaigdacaaIWaWaaWba aSqabeaacaaIXaGaaGimaaaaaaa@4A6E@

Question: 83

The speed of light in vacuum is 3× 10 8  m/s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabgE na0kaaigdacaaIWaWaaWbaaSqabeaacaaI4aaaaOGaaeiiaiaab2ga caqGVaGaae4Caaaa@3E55@ . Sunlight takes about 8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaaaa@369B@  minutes to reach the earth. Express distance of Sun from Earth in standard form.

Solution

It is given that, speed of light =3× 10 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG 4maiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaI4aaaaaaa@3C16@  m/s

Since the speed is given in m/s, we need to convert the time taken into seconds.

Time taken by light to reach the Earth =8min=8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ioaiGac2gacaGGPbGaaiOBaiabg2da9iaaiIdaaaa@3C3A@

× 60 s=480 s [ 1min=60s ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey41aqRaae iiaiaaiAdacaaIWaGaaeiiaiaadohacqGH9aqpcaaI0aGaaGioaiaa icdacaqGGaGaam4CaiaabccadaWadaqaaiablwJirjaaigdaciGGTb GaaiyAaiaac6gacqGH9aqpcaaI2aGaaGimaiaadohaaiaawUfacaGL Dbaaaaa@4B55@

We have studied that, Distance = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0daaa@36DF@  Speed × MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey41aqlaaa@37F0@  Time =3× 10 8 ×480=1440× 10 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG 4maiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacGaJaIioaaaakiab gEna0kaaisdacaaI4aGaaGimaiabg2da9iaaigdacaaI0aGaaGinai aaicdacqGHxdaTcaaIXaGaaGimamaaCaaaleqabaGaaGioaaaaaaa@49F7@

=1.440× 10 3 × 10 8 =1.44× 10 11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ymaiaac6cacaaI0aGaaGinaiaaicdacqGHxdaTcaaIXaGaaGimamaa CaaaleqabaGaaG4maaaakiabgEna0kaaigdacaaIWaWaaWbaaSqabe aacaaI4aaaaOGaeyypa0JaaGymaiaac6cacaaI0aGaaGinaiabgEna 0kaaigdacaaIWaWaaWbaaSqabeaacaaIXaGaaGymaaaaaaa@4CA4@   [  10 3 × 10 8 = 10 11 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiablw JirjaabccacaaIXaGaaGimamaaCaaaleqabaGaaG4maaaakiabgEna 0kaaigdacaaIWaWaaWbaaSqabeaacaaI4aaaaOGaeyypa0JaaGymai aaicdadaahaaWcbeqaaiaaigdacaaIXaaaaOGaaiyxaaaa@448A@

Hence, the distance of Sun from the Earth is 1.44× 10 11  m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaac6 cacaaI0aGaaGinaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaI XaGaaGymaaaakiaabccacaWGTbaaaa@3F8F@

Question: 84

Simplify and express each of the following in exponential form:

a.    [ ( 3 7 ) 4 × ( 3 7 ) 5 ]÷ ( 3 7 ) 7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada qadaqaamaalaaabaGaaG4maaqaaiaaiEdaaaaacaGLOaGaayzkaaWa aWbaaSqabeaacaaI0aaaaOGaey41aq7aaeWaaeaadaWcaaqaaiaaio daaeaacaaI3aaaaaGaayjkaiaawMcaamaaCaaaleqabaGaaGynaaaa aOGaay5waiaaw2faaiabgEpa4oaabmaabaWaaSaaaeaacaaIZaaaba GaaG4naaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiEdaaaaaaa@483A@

b.   [ ( 7 11 ) 5 ÷ ( 7 11 ) 2 ]× ( 7 11 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada qadaqaamaalaaabaGaaG4naaqaaiaaigdacaaIXaaaaaGaayjkaiaa wMcaamaaCaaaleqabaGaaGynaaaakiabgEpa4oaabmaabaWaaSaaae aacaaI3aaabaGaaGymaiaaigdaaaaacaGLOaGaayzkaaWaaWbaaSqa beaacaaIYaaaaaGccaGLBbGaayzxaaGaey41aq7aaeWaaeaadaWcaa qaaiaaiEdaaeaacaaIXaGaaGymaaaaaiaawIcacaGLPaaadaahaaWc beqaaiaaikdaaaaaaa@4A5F@

c.    ( 3 7 ÷ 3 5 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaio dadaahaaWcbeqaaiaaiEdaaaGccqGH3daUcaaIZaWaaWbaaSqabeaa caaI1aaaaOGaaiykamaaCaaaleqabaGaaGinaaaaaaa@3DBF@

d.   ( a 6 a 4 )× a 5 × a 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaadggadaahaaWcbeqaaiaaiAdaaaaakeaacaWGHbWaaWba aSqabeaacaaI0aaaaaaaaOGaayjkaiaawMcaaiabgEna0kaadggada ahaaWcbeqaaiaaiwdaaaGccqGHxdaTcaWGHbWaaWbaaSqabeaacaaI Waaaaaaa@4301@

e.    [ ( 3 5 ) 3 × ( 3 5 ) 8 ]÷[ ( 3 5 ) 2 × ( 3 5 ) 4 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada qadaqaamaalaaabaGaaG4maaqaaiaaiwdaaaaacaGLOaGaayzkaaWa aWbaaSqabeaacaaIZaaaaOGaey41aq7aaeWaaeaadaWcaaqaaiaaio daaeaacaaI1aaaaaGaayjkaiaawMcaamaaCaaaleqabaGaaGioaaaa aOGaay5waiaaw2faaiabgEpa4oaadmaabaWaaeWaaeaadaWcaaqaai aaiodaaeaacaaI1aaaaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOm aaaakiabgEna0oaabmaabaWaaSaaaeaacaaIZaaabaGaaGynaaaaai aawIcacaGLPaaadaahaaWcbeqaaiaaisdaaaaakiaawUfacaGLDbaa aaa@504F@

f. ( 5 15 ÷ 5 10 )× 5 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaiw dadaahaaWcbeqaaiaaigdacaaI1aaaaOGaey49aGRaaGynamaaCaaa leqabaGaaGymaiaaicdaaaGccaGGPaGaey41aqRaaGynamaaCaaale qabaGaaGynaaaaaaa@4209@

Solution

a.    We have, [ ( 3 7 ) 4 × ( 3 7 ) 5 ]÷ ( 3 7 ) 7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada qadaqaamaalaaabaGaaG4maaqaaiaaiEdaaaaacaGLOaGaayzkaaWa aWbaaSqabeaacaaI0aaaaOGaey41aq7aaeWaaeaadaWcaaqaaiaaio daaeaacaaI3aaaaaGaayjkaiaawMcaamaaCaaaleqabaGaaGynaaaa aOGaay5waiaaw2faaiabgEpa4oaabmaabaWaaSaaaeaacaaIZaaaba GaaG4naaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiEdaaaaaaa@483A@
= ( 3 7 ) 4+5 ÷ ( 3 7 ) 7 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaadaWcaaqaaiaaiodaaeaacaaI3aaaaaGaayjkaiaawMcaamaa CaaaleqabaGaaGinaiabgUcaRiaaiwdaaaGccqGH3daUdaqadaqaam aalaaabaGaaG4maaqaaiaaiEdaaaaacaGLOaGaayzkaaWaaWbaaSqa beaacaaI3aaaaaaa@42CC@
= ( 3 7 ) 9 ÷ ( 3 7 ) 7 = ( 3 7 ) 9 ( 3 7 ) 7 = ( 3 7 ) 97 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpdaqadaqaamaalaaabaGaaG4maaqaaiaaiEdaaaaacaGLOaGaayzk aaWaaWbaaSqabeaacaaI5aaaaOGaey49aG7aaeWaaeaadaWcaaqaai aaiodaaeaacaaI3aaaaaGaayjkaiaawMcaamaaCaaaleqabaGaaG4n aaaaaOqaaiabg2da9maalaaabaWaaeWaaeaadaWcaaqaaiaaiodaae aacaaI3aaaaaGaayjkaiaawMcaamaaCaaaleqabaGaaGyoaaaaaOqa amaabmaabaWaaSaaaeaacaaIZaaabaGaaG4naaaaaiaawIcacaGLPa aadaahaaWcbeqaaiaaiEdaaaaaaaGcbaGaeyypa0ZaaeWaaeaadaWc aaqaaiaaiodaaeaacaaI3aaaaaGaayjkaiaawMcaamaaCaaaleqaba GaaGyoaiabgkHiTiaaiEdaaaaaaaa@5132@
= ( 3 7 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaadaWcaaqaaiaaiodaaeaacaaI3aaaaaGaayjkaiaawMcaamaa CaaaleqabaGaaGOmaaaaaaa@3ADF@

b.   We have, [ ( 7 11 ) 5 ÷ ( 7 11 ) 2 ]× ( 7 11 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada qadaqaamaalaaabaGaaG4naaqaaiaaigdacaaIXaaaaaGaayjkaiaa wMcaamaaCaaaleqabaGaaGynaaaakiabgEpa4oaabmaabaWaaSaaae aacaaI3aaabaGaaGymaiaaigdaaaaacaGLOaGaayzkaaWaaWbaaSqa beaacaaIYaaaaaGccaGLBbGaayzxaaGaey41aq7aaeWaaeaadaWcaa qaaiaaiEdaaeaacaaIXaGaaGymaaaaaiaawIcacaGLPaaadaahaaWc beqaaiaaikdaaaaaaa@4A5F@
=[ ( 7 11 ) 5 ( 7 11 ) 2 ]× ( 7 11 ) 2 = ( 7 11 ) 52 × ( 7 11 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpdaWadaqaamaalaaabaWaaeWaaeaadaWcaaqaaiaaiEdaaeaacaaI XaGaaGymaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaiwdaaaaake aadaqadaqaamaalaaabaGaaG4naaqaaiaaigdacaaIXaaaaaGaayjk aiaawMcaamaaCaaaleqabaGaaGOmaaaaaaaakiaawUfacaGLDbaacq GHxdaTdaqadaqaamaalaaabaGaaG4naaqaaiaaigdacaaIXaaaaaGa ayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaOqaaiabg2da9maabm aabaWaaSaaaeaacaaI3aaabaGaaGymaiaaigdaaaaacaGLOaGaayzk aaWaaWbaaSqabeaacaaI1aGaeyOeI0IaaGOmaaaakiabgEna0oaabm aabaWaaSaaaeaacaaI3aaabaGaaGymaiaaigdaaaaacaGLOaGaayzk aaWaaWbaaSqabeaacaaIYaaaaaaaaa@578F@
= ( 7 11 ) 3 × ( 7 11 ) 2 = ( 7 11 ) 3+2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpdaqadaqaamaalaaabaGaaG4naaqaaiaaigdacaaIXaaaaaGaayjk aiaawMcaamaaCaaaleqabaGaaG4maaaakiabgEna0oaabmaabaWaaS aaaeaacaaI3aaabaGaaGymaiaaigdaaaaacaGLOaGaayzkaaWaaWba aSqabeaacGaJaIOmaaaaaOqaaiabg2da9maabmaabaWaaSaaaeaaca aI3aaabaGaaGymaiaaigdaaaaacaGLOaGaayzkaaWaaWbaaSqabeaa caaIZaGaey4kaSIaaGOmaaaaaaaa@4AF5@
= ( 7 11 ) 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaadaWcaaqaaiaaiEdaaeaacaaIXaGaaGymaaaaaiaawIcacaGL PaaadaahaaWcbeqaaiaaiwdaaaaaaa@3B9B@

c.    We have, ( 3 7 ÷ 3 5 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaio dadaahaaWcbeqaaiaaiEdaaaGccqGH3daUcaaIZaWaaWbaaSqabeaa caaI1aaaaOGaaiykamaaCaaaleqabaGaaGinaaaaaaa@3DBF@
= ( 3 75 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaai ikaiaaiodadaahaaWcbeqaaiaaiEdacqGHsislcaaI1aaaaOGaaiyk amaaCaaaleqabaGaaGinaaaaaaa@3C83@
= ( 3 2 ) 4 = 3 2×4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpdaqadaqaaiaaiodadaahaaWcbeqaaiaaikdaaaaakiaawIcacaGL PaaadaahaaWcbeqaaiaaisdaaaaakeaacqGH9aqpcaaIZaWaaWbaaS qabeaacaaIYaGaey41aqRaaGinaaaaaaaa@4093@

= 3 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG 4mamaaCaaaleqabaGaaGioaaaaaaa@388A@

d.   We have, ( a 6 a 4 )× a 5 × a 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaadggadaahaaWcbeqaaiaaiAdaaaaakeaacaWGHbWaaWba aSqabeaacaaI0aaaaaaaaOGaayjkaiaawMcaaiabgEna0kaadggada ahaaWcbeqaaiaaiwdaaaGccqGHxdaTcaWGHbWaaWbaaSqabeaacaaI Waaaaaaa@4301@
=( a 64 × a 5 ×1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaacaWGHbWaaWbaaSqabeaacaaI2aGaeyOeI0IaaGinaaaakiab gEna0kaadggadaahaaWcbeqaaiaaiwdaaaGccqGHxdaTcaaIXaaaca GLOaGaayzkaaaaaa@42B5@

= a 2 × a 5 = a 2+5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaWGHbWaaWbaaSqabeaacaaIYaaaaOGaey41aqRaamyyamaaCaaa leqabaGaaGynaaaaaOqaaiabg2da9iaadggadaahaaWcbeqaaiaaik dacqGHRaWkcaaI1aaaaaaaaa@4127@
= a 7 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam yyamaaCaaaleqabaGaaG4naaaaaaa@38B3@

e.    We have, [ ( 3 5 ) 3 × ( 3 5 ) 8 ]÷[ ( 3 5 ) 2 × ( 3 5 ) 4 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada qadaqaamaalaaabaGaaG4maaqaaiaaiwdaaaaacaGLOaGaayzkaaWa aWbaaSqabeaacaaIZaaaaOGaey41aq7aaeWaaeaadaWcaaqaaiaaio daaeaacaaI1aaaaaGaayjkaiaawMcaamaaCaaaleqabaGaaGioaaaa aOGaay5waiaaw2faaiabgEpa4oaadmaabaWaaeWaaeaadaWcaaqaai aaiodaaeaacaaI1aaaaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOm aaaakiabgEna0oaabmaabaWaaSaaaeaacaaIZaaabaGaaGynaaaaai aawIcacaGLPaaadaahaaWcbeqaaiaaisdaaaaakiaawUfacaGLDbaa aaa@504F@
= ( 3 5 ) 3+8 ÷ ( 3 5 ) 2+4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaadaWcaaqaaiaaiodaaeaacaaI1aaaaaGaayjkaiaawMcaamaa CaaaleqabaGaaG4maiabgUcaRiaaiIdaaaGccqGH3daUdaqadaqaam aalaaabaGaaG4maaqaaiaaiwdaaaaacaGLOaGaayzkaaWaaWbaaSqa beaacaaIYaGaey4kaSIaaGinaaaaaaa@4465@
= ( 3 5 ) 11 ÷ ( 3 5 ) 6 = ( 3 5 ) 11 ( 3 5 ) 6 = ( 3 5 ) 116 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpdaqadaqaamaalaaabaGaaG4maaqaaiaaiwdaaaaacaGLOaGaayzk aaWaaWbaaSqabeaacaaIXaGaaGymaaaakiabgEpa4oaabmaabaWaaS aaaeaacaaIZaaabaGaaGynaaaaaiaawIcacaGLPaaadaahaaWcbeqa aiaaiAdaaaaakeaacqGH9aqpdaWcaaqaamaabmaabaWaaSaaaeaaca aIZaaabaGaaGynaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaigda caaIXaaaaaGcbaWaaeWaaeaadaWcaaqaaiaaiodaaeaacaaI1aaaaa GaayjkaiaawMcaamaaCaaaleqabaGaaGOnaaaaaaaakeaacqGH9aqp daqadaqaamaalaaabaGaaG4maaqaaiaaiwdaaaaacaGLOaGaayzkaa WaaWbaaSqabeaacaaIXaGaaGymaiabgkHiTiaaiAdaaaaaaaa@533D@
= ( 3 5 ) 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaadaWcaaqaaiaaiodaaeaacaaI1aaaaaGaayjkaiaawMcaamaa CaaaleqabaGaaGynaaaaaaa@3AE0@

f.We have, ( 5 15 ÷ 5 10 )× 5 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aI1aWaaWbaaSqabeaacaaIXaGaaGynaaaakiabgEpa4kaaiwdadaah aaWcbeqaaiaaigdacaaIWaaaaaGccaGLOaGaayzkaaGaey41aqRaaG ynamaaCaaaleqabaGaaGynaaaaaaa@423A@
=( 5 15 5 10 )× 5 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaadaWcaaqaaiaaiwdadaahaaWcbeqaaiaaigdacaaI1aaaaaGc baGaaGynamaaCaaaleqabaGaaGymaiaaicdaaaaaaaGccaGLOaGaay zkaaGaey41aqRaaGynamaaCaaaleqabaGaaGynaaaaaaa@4115@

= 5 5 × 5 5 = 5 5+5 = 5 10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaaI1aWaaWbaaSqabeaacaaI1aaaaOGaey41aqRaaGynamaaCaaa leqabaGaaGynaaaaaOqaaiabg2da9iaaiwdadaahaaWcbeqaaiaaiw dacqGHRaWkcaaI1aaaaaGcbaGaeyypa0JaaGynamaaCaaaleqabaGa aGymaiaaicdaaaaaaaa@442A@

Question: 85

Evaluate

a.    7 8 × a 10 b 7 c 12 7 6 × a 8 b 4 c 12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI3aWaaWbaaSqabeaacaaI4aaaaOGaey41aqRaamyyamaaCaaaleqa baGaaGymaiaaicdaaaGccaWGIbWaaWbaaSqabeaacaaI3aaaaOGaam 4yamaaCaaaleqabaGaaGymaiaaikdaaaaakeaacaaI3aWaaWbaaSqa beaacaaI2aaaaOGaey41aqRaamyyamaaCaaaleqabaGaaGioaaaaki aadkgadaahaaWcbeqaaiaaisdaaaGccaWGJbWaaWbaaSqabeaacaaI XaGaaGOmaaaaaaaaaa@4AD7@

b.   5 4 × 7 4 × 2 7 8×49× 5 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI1aWaaWbaaSqabeaacaaI0aaaaOGaey41aqRaaG4namaaCaaaleqa baGaaGinaaaakiabgEna0kaaikdadaahaaWcbeqaaiaaiEdaaaaake aacaaI4aGaey41aqRaaGinaiaaiMdacqGHxdaTcaaI1aWaaWbaaSqa beaacaaIZaaaaaaaaaa@474F@

c.    125× 5 2 × a 7 10 3 × a 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaGaaGOmaiaaiwdacqGHxdaTcaaI1aWaaWbaaSqabeaacaaIYaaa aOGaey41aqRaamyyamaaCaaaleqabaGaaG4naaaaaOqaaiaaigdaca aIWaWaaWbaaSqabeaacaaIZaaaaOGaey41aqRaamyyamaaCaaaleqa baGaaGinaaaaaaaaaa@462E@

d.   3 4 × 12 3 ×36 2 5 × 6 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaWaaWbaaSqabeaacaaI0aaaaOGaey41aqRaaGymaiaaikdadaah aaWcbeqaaiaaiodaaaGccqGHxdaTcaaIZaGaaGOnaaqaaiaaikdada ahaaWcbeqaaiaaiwdaaaGccqGHxdaTcaaI2aWaaWbaaSqabeaacaaI Zaaaaaaaaaa@4524@

e.    ( 6×10 2 2 × 5 3 ) 2 × 25 27 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaiAdacqGHxdaTcaaIXaGaaGimaaqaaiaaikdadaahaaWc beqaaiaaikdaaaGccqGHxdaTcaaI1aWaaWbaaSqabeaacaaIZaaaaa aaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaakiabgEna0oaa laaabaGaaGOmaiaaiwdaaeaacaaIYaGaaG4naaaaaaa@4749@

f. 15 4 × 18 3 3 3 × 5 2 × 12 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaGaaGynamaaCaaaleqabaGaaGinaaaakiabgEna0kaaigdacaaI 4aWaaWbaaSqabeaacaaIZaaaaaGcbaGaaG4mamaaCaaaleqabaGaaG 4maaaakiabgEna0kaaiwdadaahaaWcbeqaaiaaikdaaaGccqGHxdaT caaIXaGaaGOmamaaCaaaleqabaGaaGOmaaaaaaaaaa@46D1@

g.   6 4 × 9 2 × 25 3 3 2 × 4 2 × 15 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI2aWaaWbaaSqabeaacaaI0aaaaOGaey41aqRaaGyoamaaCaaaleqa baGaaGOmaaaakiabgEna0kaaikdacaaI1aWaaWbaaSqabeaacaaIZa aaaaGcbaGaaG4mamaaCaaaleqabaGaaGOmaaaakiabgEna0kaaisda daahaaWcbeqaaiaaikdaaaGccqGHxdaTcaaIXaGaaGynamaaCaaale qabaGaaGOnaaaaaaaaaa@49E7@

Solution

a.    Considering the expression, 7 8 × a 10 b 7 c 12 7 6 × a 8 b 4 c 12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI3aWaaWbaaSqabeaacaaI4aaaaOGaey41aqRaamyyamaaCaaaleqa baGaaGymaiaaicdaaaGccaWGIbWaaWbaaSqabeaacaaI3aaaaOGaam 4yamaaCaaaleqabaGaaGymaiaaikdaaaaakeaacaaI3aWaaWbaaSqa beaacaaI2aaaaOGaey41aqRaamyyamaaCaaaleqabaGaaGioaaaaki aadkgadaahaaWcbeqaaiaaisdaaaGccaWGJbWaaWbaaSqabeaacaaI XaGaaGOmaaaaaaaaaa@4AD7@

=( 7 8 7 6 )×( a 10 a 8 )×( b 7 b 4 )×( c 12 c 12 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaadaWcaaqaaiaaiEdadaahaaWcbeqaaiaaiIdaaaaakeaacaaI 3aWaaWbaaSqabeaacaaI2aaaaaaaaOGaayjkaiaawMcaaiabgEna0o aabmaabaWaaSaaaeaacaWGHbWaaWbaaSqabeaacaaIXaGaaGimaaaa aOqaaiaadggadaahaaWcbeqaaiaaiIdaaaaaaaGccaGLOaGaayzkaa Gaey41aq7aaeWaaeaadaWcaaqaaiaadkgadaahaaWcbeqaaiaaiEda aaaakeaacaWGIbWaaWbaaSqabeaacaaI0aaaaaaaaOGaayjkaiaawM caaiabgEna0oaabmaabaWaaSaaaeaacaWGJbWaaWbaaSqabeaacaaI XaGaaGOmaaaaaOqaaiaadogadaahaaWcbeqaaiaaigdacaaIYaaaaa aaaOGaayjkaiaawMcaaaaa@5452@

= 7 86 × a 108 × b 74 × c 1212 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG 4namaaCaaaleqabaGaaGioaiabgkHiTiaaiAdaaaGccqGHxdaTcaWG HbWaaWbaaSqabeaacaaIXaGaaGimaiabgkHiTiaaiIdaaaGccqGHxd aTcaWGIbWaaWbaaSqabeaacaaI3aGaeyOeI0IaaGinaaaakiabgEna 0kaadogadaahaaWcbeqaaiaaigdacaaIYaGaeyOeI0IaaGymaiaaik daaaaaaa@4D45@

= 7 2 × a 2 × b 3 × c 0 =49 a 2 b 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaaI3aWaaWbaaSqabeaacaaIYaaaaOGaey41aqRaamyyamaaCaaa leqabaGaaGOmaaaakiabgEna0kaadkgadaahaaWcbeqaaiaaiodaaa GccqGHxdaTcaWGJbWaaWbaaSqabeaacaaIWaaaaaGcbaGaeyypa0Ja aGinaiaaiMdacaWGHbWaaWbaaSqabeaacaaIYaaaaOGaamOyamaaCa aaleqabaGaaG4maaaaaaaa@4A9C@

b.      Considering the expression, 5 4 × 7 4 × 2 7 8×49× 5 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI1aWaaWbaaSqabeaacaaI0aaaaOGaey41aqRaaG4namaaCaaaleqa baGaaGinaaaakiabgEna0kaaikdadaahaaWcbeqaaiaaiEdaaaaake aacaaI4aGaey41aqRaaGinaiaaiMdacqGHxdaTcaaI1aWaaWbaaSqa beaacaaIZaaaaaaaaaa@474E@

= 5 4 × 7 4 × 2 7 2 3 × 7 2 × 5 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS aaaeaacaaI1aWaaWbaaSqabeaacaaI0aaaaOGaey41aqRaaG4namaa CaaaleqabaGaaGinaaaakiabgEna0kaaikdadaahaaWcbeqaaiaaiE daaaaakeaacaaIYaWaaWbaaSqabeaacaaIZaaaaOGaey41aqRaaG4n amaaCaaaleqabaGaaGOmaaaakiabgEna0kaaiwdadaahaaWcbeqaai aaiodaaaaaaaaa@4976@

=( 5 4 5 3 )×( 2 7 2 3 )×( 7 4 7 2 ) = 5 43 × 2 73 × 7 42 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpdaqadaqaamaalaaabaGaaGynamaaCaaaleqabaGaaGinaaaaaOqa aiaaiwdadaahaaWcbeqaaiaaiodaaaaaaaGccaGLOaGaayzkaaGaey 41aq7aaeWaaeaadaWcaaqaaiaaikdadaahaaWcbeqaaiaaiEdaaaaa keaacaaIYaWaaWbaaSqabeaacaaIZaaaaaaaaOGaayjkaiaawMcaai abgEna0oaabmaabaWaaSaaaeaacaaI3aWaaWbaaSqabeaacaaI0aaa aaGcbaGaaG4namaaCaaaleqabaGaaGOmaaaaaaaakiaawIcacaGLPa aaaeaacqGH9aqpcaaI1aWaaWbaaSqabeaacaaI0aGaeyOeI0IaaG4m aaaakiabgEna0kaaikdadaahaaWcbeqaaiaaiEdacqGHsislcaaIZa aaaOGaey41aqRaaG4namaaCaaaleqabaGaaGinaiabgkHiTiaaikda aaaaaaa@5959@

=5× 2 4 × 7 2 =5×16×49 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaaI1aGaey41aqRaaGOmamaaCaaaleqabaGaaGinaaaakiabgEna 0kaaiEdadaahaaWcbeqaaiaaikdaaaaakeaacqGH9aqpcaaI1aGaey 41aqRaaGymaiaaiAdacqGHxdaTcaaI0aGaaGyoaaaaaa@4827@

=3920 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG 4maiaaiMdacaaIYaGaaGimaaaa@39D5@

c.    Considering the expression, 125× 5 2 × a 7 10 3 × a 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaGaaGOmaiaaiwdacqGHxdaTcaaI1aWaaWbaaSqabeaacaaIYaaa aOGaey41aqRaamyyamaaCaaaleqabaGaaG4naaaaaOqaaiaaigdaca aIWaWaaWbaaSqabeaacaaIZaaaaOGaey41aqRaamyyamaaCaaaleqa baGaaGinaaaaaaaaaa@462E@

= 5 3 × 5 2 × a 7 ( 2×5 ) 3 × a 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS aaaeaacaaI1aWaaWbaaSqabeaacaaIZaaaaOGaey41aqRaaGynamaa CaaaleqabaGaaGOmaaaakiabgEna0kaadggadaahaaWcbeqaaiaaiE daaaaakeaadaqadaqaaiaaikdacqGHxdaTcaaI1aaacaGLOaGaayzk aaWaaWbaaSqabeaacaaIZaaaaOGaey41aqRaamyyamaaCaaaleqaba GaaGinaaaaaaaaaa@4A57@

= 5 3+2 × a 7 2 3 × 5 3 × a 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS aaaeaacaaI1aWaaWbaaSqabeaacaaIZaGaey4kaSIaaGOmaaaakiab gEna0kaadggadaahaaWcbeqaaiaaiEdaaaaakeaacaaIYaWaaWbaaS qabeaacaaIZaaaaOGaey41aqRaaGynamaaCaaaleqabaGaaG4maaaa kiabgEna0kaadggadaahaaWcbeqaaiaaisdaaaaaaaaa@4797@

= 5 5 × a 7 2 3 × 5 3 × a 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS aaaeaacaaI1aWaaWbaaSqabeaacaaI1aaaaOGaey41aqRaamyyamaa CaaaleqabaGaaG4naaaaaOqaaiaaikdadaahaaWcbeqaaiaaiodaaa GccqGHxdaTcaaI1aWaaWbaaSqabeaacaaIZaaaaOGaey41aqRaamyy amaaCaaaleqabaGaaGinaaaaaaaaaa@45FB@

=( 5 5 5 3 )×( a 7 a 4 )×( 1 2 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaadaWcaaqaaiaaiwdadaahaaWcbeqaaiaaiwdaaaaakeaacaaI 1aWaaWbaaSqabeaacaaIZaaaaaaaaOGaayjkaiaawMcaaiabgEna0o aabmaabaWaaSaaaeaacaWGHbWaaWbaaSqabeaacaaI3aaaaaGcbaGa amyyamaaCaaaleqabaGaaGinaaaaaaaakiaawIcacaGLPaaacqGHxd aTdaqadaqaamaalaaabaGaaGymaaqaaiaaikdadaahaaWcbeqaaiaa iodaaaaaaaGccaGLOaGaayzkaaaaaa@4964@

=( 5 53 × a 74 2 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaadaWcaaqaaiaaiwdadaahaaWcbeqaaiaaiwdacqGHsislcaaI ZaaaaOGaey41aqRaamyyamaaCaaaleqabaGaaG4naiabgkHiTiaais daaaaakeaacaaIYaWaaWbaaSqabeaacaaIZaaaaaaaaOGaayjkaiaa wMcaaaaa@4327@

= 5 2 × a 3 2 3 = 25 a 3 8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpdaWcaaqaaiaaiwdadaahaaWcbeqaaiaaikdaaaGccqGHxdaTcaWG HbWaaWbaaSqabeaacaaIZaaaaaGcbaGaaGOmamaaCaaaleqabaGaaG 4maaaaaaaakeaacqGH9aqpdaWcaaqaaiaaikdacaaI1aGaamyyamaa CaaaleqabaGaaG4maaaaaOqaaiaaiIdaaaaaaaa@4375@

d.   Considering the expression, 3 4 × 12 3 ×36 2 5 × 6 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaWaaWbaaSqabeaacaaI0aaaaOGaey41aqRaaGymaiaaikdadaah aaWcbeqaaiaaiodaaaGccqGHxdaTcaaIZaGaaGOnaaqaaiaaikdada ahaaWcbeqaaiaaiwdaaaGccqGHxdaTcaaI2aWaaWbaaSqabeaacaaI Zaaaaaaaaaa@4524@

= 3 4 ×( 2 6 × 3 3 )×( 2 2 × 3 2 ) 2 5 × ( 2×3 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS aaaeaacaaIZaWaaWbaaSqabeaacaaI0aaaaOGaey41aq7aaeWaaeaa caaIYaWaaWbaaSqabeaacaaI2aaaaOGaey41aqRaaG4mamaaCaaale qabaGaaG4maaaaaOGaayjkaiaawMcaaiabgEna0oaabmaabaGaaGOm amaaCaaaleqabaGaaGOmaaaakiabgEna0kaaiodadaahaaWcbeqaai aaikdaaaaakiaawIcacaGLPaaaaeaacaaIYaWaaWbaaSqabeaacaaI 1aaaaOGaey41aq7aaeWaaeaacaaIYaGaey41aqRaaG4maaGaayjkai aawMcaamaaCaaaleqabaGaaG4maaaaaaaaaa@549D@

= 3 4 × 2 6 × 3 3 × 2 2 × 3 2 2 5 × 2 3 × 3 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS aaaeaacaaIZaWaaWbaaSqabeaacaaI0aaaaOGaey41aqRaaGOmamaa CaaaleqabaGaaGOnaaaakiabgEna0kaaiodadaahaaWcbeqaaiaaio daaaGccqGHxdaTcaaIYaWaaWbaaSqabeaacaaIYaaaaOGaey41aqRa aG4mamaaCaaaleqabaGaaGOmaaaaaOqaaiaaikdadaahaaWcbeqaai aaiwdaaaGccqGHxdaTcaaIYaWaaWbaaSqabeaacaaIZaaaaOGaey41 aqRaaG4mamaaCaaaleqabaGaaG4maaaaaaaaaa@50F7@

( 3 4 × 3 3 × 3 2 )×( 2 6 × 2 2 ) 2 5 × 2 3 × 3 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaada qadaqaaiaaiodadaahaaWcbeqaaiaaisdaaaGccqGHxdaTcaaIZaWa aWbaaSqabeaacaaIZaaaaOGaey41aqRaaG4mamaaCaaaleqabaGaaG OmaaaaaOGaayjkaiaawMcaaiabgEna0oaabmaabaGaaGOmamaaCaaa leqabaGaaGOnaaaakiabgEna0kaaikdadaahaaWcbeqaaiaaikdaaa aakiaawIcacaGLPaaaaeaacaaIYaWaaWbaaSqabeaacaaI1aaaaOGa ey41aqRaaGOmamaaCaaaleqabaGaaG4maaaakiabgEna0kaaiodada ahaaWcbeqaaiaaiodaaaaaaaaa@5302@

= 3 4+3+2 × 2 6+2 2 5+3 × 3 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS aaaeaacaaIZaWaaWbaaSqabeaacaaI0aGaey4kaSIaaG4maiabgUca RiaaikdaaaGccqGHxdaTcaaIYaWaaWbaaSqabeaacaaI2aGaey4kaS IaaGOmaaaaaOqaaiaaikdadaahaaWcbeqaaiaaiwdacqGHRaWkcaaI ZaaaaOGaey41aqRaaG4mamaaCaaaleqabaGaaG4maaaaaaaaaa@4855@

= 3 9 × 2 8 3 3 × 2 8 = 3 93 × 2 88 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS aaaeaacaaIZaWaaWbaaSqabeaacaaI5aaaaOGaey41aqRaaGOmamaa CaaaleqabaGaaGioaaaaaOqaaiaaiodadaahaaWcbeqaaiaaiodaaa GccqGHxdaTcaaIYaWaaWbaaSqabeaacaaI4aaaaaaakiabg2da9iaa iodadaahaaWcbeqaaiaaiMdacqGHsislcaaIZaaaaOGaey41aqRaaG OmamaaCaaaleqabaGaaGioaiabgkHiTiaaiIdaaaaaaa@4BC7@

= 3 6 × 2 0 = 3 6 ×1 =729 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaaIZaWaaWbaaSqabeaacaaI2aaaaOGaey41aqRaaGOmamaaCaaa leqabaGaaGimaaaakiabg2da9iaaiodadaahaaWcbeqaaiaaiAdaaa GccqGHxdaTcaaIXaaabaGaeyypa0JaaG4naiaaikdacaaI5aaaaaa@4530@

e.    Considering the expression, ( 6×10 2 2 × 5 3 ) 2 × 25 27 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaiAdacqGHxdaTcaaIXaGaaGimaaqaaiaaikdadaahaaWc beqaaiaaikdaaaGccqGHxdaTcaaI1aWaaWbaaSqabeaacaaIZaaaaa aaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaakiabgEna0oaa laaabaGaaGOmaiaaiwdaaeaacaaIYaGaaG4naaaaaaa@4749@

= ( 2×3×2×5 2 2 × 5 3 ) 2 × 5 2 3 3 = ( 3 5 2 ) 2 × 5 2 3 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpdaqadaqaamaalaaabaGaaGOmaiabgEna0kaaiodacqGHxdaTcaaI YaGaey41aqRaaGynaaqaaiaaikdadaahaaWcbeqaaiaaikdaaaGccq GHxdaTcaaI1aWaaWbaaSqabeaacaaIZaaaaaaaaOGaayjkaiaawMca amaaCaaaleqabaGaaGOmaaaakiabgEna0oaalaaabaGaaGynamaaCa aaleqabaGaaGOmaaaaaOqaaiaaiodadaahaaWcbeqaaiaaiodaaaaa aaGcbaGaeyypa0ZaaeWaaeaadaWcaaqaaiaaiodaaeaacaaI1aWaaW baaSqabeaacaaIYaaaaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGa aGOmaaaakiabgEna0oaalaaabaGaaGynamaaCaaaleqabaGaaGOmaa aaaOqaaiaaiodadaahaaWcbeqaaiaaiodaaaaaaaaaaa@592E@

= 3 2 5 4 × 5 2 3 3 = 1 5 2 ×3 = 1 25×3 = 1 75 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpdaWcaaqaaiaaiodadaahaaWcbeqaaiaaikdaaaaakeaacaaI1aWa aWbaaSqabeaacaaI0aaaaaaakiabgEna0oaalaaabaGaaGynamaaCa aaleqabaGaaGOmaaaaaOqaaiaaiodadaahaaWcbeqaaiaaiodaaaaa aaGcbaGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGynamaaCaaaleqaba GaaGOmaaaakiabgEna0kaaiodaaaaabaGaeyypa0ZaaSaaaeaacaaI XaaabaGaaGOmaiaaiwdacqGHxdaTcaaIZaaaaaqaaiabg2da9maala aabaGaaGymaaqaaiaaiEdacaaI1aaaaaaaaa@4FAD@

f.     Considering the expression,, 15 4 × 18 3 3 3 × 5 2 × 12 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaGaaGynamaaCaaaleqabaGaaGinaaaakiabgEna0kaaigdacaaI 4aWaaWbaaSqabeaacaaIZaaaaaGcbaGaaG4mamaaCaaaleqabaGaaG 4maaaakiabgEna0kaaiwdadaahaaWcbeqaaiaaikdaaaGccqGHxdaT caaIXaGaaGOmamaaCaaaleqabaGaaGOmaaaaaaaaaa@46D1@

= ( 3×5 ) 4 × ( 2× 3 2 ) 3 3 3 × 5 2 × ( 2 2 ×3 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS aaaeaadaqadaqaaiaaiodacqGHxdaTcaaI1aaacaGLOaGaayzkaaWa aWbaaSqabeaacaaI0aaaaOGaey41aq7aaeWaaeaacaaIYaGaey41aq RaaG4mamaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaamaaCaaa leqabaGaaG4maaaaaOqaaiaaiodadaahaaWcbeqaaiaaiodaaaGccq GHxdaTcaaI1aWaaWbaaSqabeaacaaIYaaaaOGaey41aq7aaeWaaeaa caaIYaWaaWbaaSqabeaacaaIYaaaaOGaey41aqRaaG4maaGaayjkai aawMcaamaaCaaaleqabaGaaGOmaaaaaaaaaa@549D@

= 3 4+632 × 5 4 2 2 43 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS aaaeaacaaIZaWaaWbaaSqabeaacaaI0aGaey4kaSIaaGOnaiabgkHi TiaaiodacqGHsislcaaIYaaaaOGaey41aqRaaGynamaaCaaaleqaba GaaGinaaaakmaaCaaaleqabaGaeyOeI0IaaGOmaaaaaOqaaiaaikda daahaaWcbeqaaiaaisdacqGHsislcaaIZaaaaaaaaaa@4692@

= 3 5 × 5 2 2 = ( 243×25 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpdaWcaaqaaiaaiodadaahaaWcbeqaaiaaiwdaaaGccqGHxdaTcaaI 1aWaaWbaaSqabeaacaaIYaaaaaGcbaGaaGOmaaaaaeaacqGH9aqpda WcaaqaamaabmaabaGaaGOmaiaaisdacaaIZaGaey41aqRaaGOmaiaa iwdaaiaawIcacaGLPaaaaeaacaaIYaaaaaaaaa@4651@

= 6075 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS aaaeaacaaI2aGaaGimaiaaiEdacaaI1aaabaGaaGOmaaaaaaa@3AA5@

g.   Considering the expression,, = 6 4 × 9 2 × 25 3 3 2 × 4 2 × 15 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS aaaeaacaaI2aWaaWbaaSqabeaacaaI0aaaaOGaey41aqRaaGyoamaa CaaaleqabaGaaGOmaaaakiabgEna0kaaikdacaaI1aWaaWbaaSqabe aacaaIZaaaaaGcbaGaaG4mamaaCaaaleqabaGaaGOmaaaakiabgEna 0kaaisdadaahaaWcbeqaaiaaikdaaaGccqGHxdaTcaaIXaGaaGynam aaCaaaleqabaGaaGOnaaaaaaaaaa@4AED@

= ( 2×3 ) 4 × ( 3 2 ) 2 × ( 5 2 ) 3 3 2 × 2 4 × 3 6 × 5 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS aaaeaadaqadaqaaiaaikdacqGHxdaTcaaIZaaacaGLOaGaayzkaaWa aWbaaSqabeaacaaI0aaaaOGaey41aq7aaeWaaeaacaaIZaWaaWbaaS qabeaacaaIYaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaa aOGaey41aq7aaeWaaeaacaaI1aWaaWbaaSqabeaacaaIYaaaaaGcca GLOaGaayzkaaWaaWbaaSqabeaacaaIZaaaaaGcbaGaaG4mamaaCaaa leqabaGaaGOmaaaakiabgEna0kaaikdadaahaaWcbeqaaiaaisdaaa GccqGHxdaTcaaIZaWaaWbaaSqabeaacaaI2aaaaOGaey41aqRaaGyn amaaCaaaleqabaGaaGOnaaaaaaaaaa@568C@

= 2 44 × 3 4+426 × 5 66 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG OmamaaCaaaleqabaGaaGinaiabgkHiTiaaisdaaaGccqGHxdaTcaaI ZaWaaWbaaSqabeaacaaI0aGaey4kaSIaaGinaiabgkHiTiaaikdacq GHsislcaaI2aaaaOGaey41aqRaaGynamaaCaaaleqabaGaaGOnaiab gkHiTiaaiAdaaaaaaa@486A@

= 2 0 × 3 0 × 5 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG OmamaaCaaaleqabaGaaGimaaaakiabgEna0kaaiodadaahaaWcbeqa aiaaicdaaaGccqGHxdaTcaaI1aWaaWbaaSqabeaacaaIWaaaaaaa@400E@

=1×1×1 =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaaIXaGaey41aqRaaGymaiabgEna0kaaigdaaeaacqGH9aqpcaaI Xaaaaaa@3F05@

Question: 86

Express the given information in Scientific notation (standard form) and then arrange them in ascending order of their size.

Sl.No.

Deserts of the World

Area (Sq. Kilometres)

(1)

Kalahari, South Africa

932,400 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoaiaaio dacaaIYaGaaiilaiaaisdacaaIWaGaaGimaaaa@3AF7@

(2)

Thar, India

199,430 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaiM dacaaI5aGaaiilaiaaisdacaaIZaGaaGimaaaa@3AFF@

(3)

Gibson, Australia

155,400 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaiw dacaaI1aGaaiilaiaaisdacaaIWaGaaGimaaaa@3AF4@

(4)

Great Victoria, Australia

647,500 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnaiaais dacaaI3aGaaiilaiaaiwdacaaIWaGaaGimaaaa@3AFB@

(5)

Sahara, North Africa

8,598,800 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaacY cacaaI1aGaaGyoaiaaiIdacaGGSaGaaGioaiaaicdacaaIWaaaaa@3C75@

Solution

1.   Area of Kalahari, South Africa   

=932,400=932400.00 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG yoaiaaiodacaaIYaGaaiilaiaaisdacaaIWaGaaGimaiabg2da9iaa iMdacaaIZaGaaGOmaiaaisdacaaIWaGaaGimaiaac6cacaaIWaGaaG imaaaa@4397@  

[Since we know that the standard form is written as a× 10 k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgE na0kaaigdacaaIWaWaaWbaaSqabeaacaWGRbaaaaaa@3B68@  ]

     =9324× 10 2 =9.324× 10 3 × 10 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaaI5aGaaG4maiaaikdacaaI0aGaey41aqRaaGymaiaaicdadaah aaWcbeqaaiaaikdaaaaakeaacqGH9aqpcaaI5aGaaiOlaiaaiodaca aIYaGaaGinaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaIZaaa aOGaey41aqRaaGymaiaaicdadaahaaWcbeqaaiaaikdaaaaaaaa@4C05@  

     =9.324× 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG yoaiaac6cacaaIZaGaaGOmaiaaisdacqGHxdaTcaaIXaGaaGimamaa CaaaleqabaGaaGynaaaaaaa@3F03@

2.   Area of Thar, India =199,430=199430.00 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ymaiaaiMdacaaI5aGaaiilaiaaisdacaaIZaGaaGimaiabg2da9iaa igdacaaI5aGaaGyoaiaaisdacaaIZaGaaGimaiaac6cacaaIWaGaaG imaaaa@43A7@

=19943× 10 1 =1.9943× 10 4 × 10 1 =1.9943× 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaaIXaGaaGyoaiaaiMdacaaI0aGaaG4maiabgEna0kaaigdacaaI WaWaaWbaaSqabeaacaaIXaaaaaGcbaGaeyypa0JaaGymaiaac6caca aI5aGaaGyoaiaaisdacaaIZaGaey41aqRaaGymaiaaicdadaahaaWc beqaaiaaisdaaaGccqGHxdaTcaaIXaGaaGimamaaCaaaleqabaGaaG ymaaaaaOqaaiabg2da9iaaigdacaGGUaGaaGyoaiaaiMdacaaI0aGa aG4maiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaI1aaaaaaaaa@5780@

3.   Area of Gibson, Australia =155,400 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ymaiaaiwdacaaI1aGaaiilaiaaisdacaaIWaGaaGimaaaa@3BFA@

=155400.00 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ymaiaaiwdacaaI1aGaaGinaiaaicdacaaIWaGaaiOlaiaaicdacaaI Waaaaa@3D70@

=1554× 10 2 =1.554× 10 3 × 10 2 =1.554× 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaaIXaGaaGynaiaaiwdacaaI0aGaey41aqRaaGymaiaaicdadaah aaWcbeqaaiaaikdaaaaakeaacqGH9aqpcaaIXaGaaiOlaiaaiwdaca aI1aGaaGinaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaIZaaa aOGaey41aqRaaGymaiaaicdadaahaaWcbeqaaiaaikdaaaaakeaacq GH9aqpcaaIXaGaaiOlaiaaiwdacaaI1aGaaGinaiabgEna0kaaigda caaIWaWaaWbaaSqabeaacaaI1aaaaaaaaa@5532@

4.   Area of Great Victoria, Australia MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcaa@35D8@ =647,500 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG OnaiaaisdacaaI3aGaaiilaiaaiwdacaaIWaGaaGimaaaa@3C01@

=6475× 10 2 =6.475× 10 3 × 10 5 =6.475× 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaaI2aGaaGinaiaaiEdacaaI1aGaey41aqRaaGymaiaaicdadaah aaWcbeqaaiaaikdaaaaakeaacqGH9aqpcaaI2aGaaiOlaiaaisdaca aI3aGaaGynaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacaaIZaaa aOGaey41aqRaaGymaiaaicdadaahaaWcbeqaaiaaiwdaaaaakeaacq GH9aqpcaaI2aGaaiOlaiaaisdacaaI3aGaaGynaiabgEna0kaaigda caaIWaWaaWbaaSqabeaacaaI1aaaaaaaaa@554A@

5.   Area of Sahara, North-Africa =8,598,800 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ioaiaacYcacaaI1aGaaGyoaiaaiIdacaGGSaGaaGioaiaaicdacaaI Waaaaa@3D7B@

=85988× 10 2 =8.5988× 10 4 × 10 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaaI4aGaaGynaiaaiMdacaaI4aGaaGioaiabgEna0kaaigdacaaI WaWaaWbaaSqabeaacaaIYaaaaaGcbaGaeyypa0JaaGioaiaac6caca aI1aGaaGyoaiaaiIdacaaI4aGaey41aqRaaGymaiaaicdadaahaaWc beqaaiaaisdaaaGccqGHxdaTcaaIXaGaaGimamaaCaaaleqabaGaaG Omaaaaaaaa@4DA2@

=8.5988× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ioaiaac6cacaaI1aGaaGyoaiaaiIdacaaI4aGaey41aqRaaGymaiaa icdadaahaaWcbeqaaiaaiAdaaaaaaa@3FD2@

We have studied that, to compare two numbers written in scientific notation: The number with the larger power of 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic daaaa@374E@  is greater than the number with the smaller power of 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic daaaa@374E@ . If the powers of ten are the same, then the number with larger factor is the larger number. Hence, the ascending order of the sizes of the deserts will be Gibson, Australia < MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGH8aapaaa@36FC@  Thar, India < MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGH8aapaaa@36FC@  Great Victoria, Australia < MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGH8aapaaa@36FC@  Kalahari, South-Africa < MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGH8aapaaa@36FC@  Sahara, North-Africa.

Question: 87

Express the given information in Scientific notation and then arrange them in descending order of their size.

Sl.No.

Name of the Planet

Mass (in kg)

(1)

Mercury

330000000000000000000000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaaio dacaaIWaGaaGimaiaaicdacaaIWaGaaGimaiaaicdacaaIWaGaaGim aiaaicdacaaIWaGaaGimaiaaicdacaaIWaGaaGimaiaaicdacaaIWa GaaGimaiaaicdacaaIWaGaaGimaiaaicdacaaIWaaaaa@474E@

(2)

Venus

4870000000000000000000000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiaaiI dacaaI3aGaaGimaiaaicdacaaIWaGaaGimaiaaicdacaaIWaGaaGim aiaaicdacaaIWaGaaGimaiaaicdacaaIWaGaaGimaiaaicdacaaIWa GaaGimaiaaicdacaaIWaGaaGimaiaaicdacaaIWaGaaGimaaaa@4815@

(3)

Earth

5980000000000000000000000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaiaaiM dacaaI4aGaaGimaiaaicdacaaIWaGaaGimaiaaicdacaaIWaGaaGim aiaaicdacaaIWaGaaGimaiaaicdacaaIWaGaaGimaiaaicdacaaIWa GaaGimaiaaicdacaaIWaGaaGimaiaaicdacaaIWaGaaGimaaaa@4818@

(4)

Mars

642000000000000000000000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnaiaais dacaaIYaGaaGimaiaaicdacaaIWaGaaGimaiaaicdacaaIWaGaaGim aiaaicdacaaIWaGaaGimaiaaicdacaaIWaGaaGimaiaaicdacaaIWa GaaGimaiaaicdacaaIWaGaaGimaiaaicdacaaIWaaaaa@4754@

(5)

Jupiter

190000000000000000000000 0000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaaIXa GaaGyoaiaaicdacaaIWaGaaGimaiaaicdacaaIWaGaaGimaiaaicda caaIWaGaaGimaiaaicdacaaIWaGaaGimaiaaicdacaaIWaGaaGimai aaicdacaaIWaGaaGimaiaaicdacaaIWaGaaGimaiaaicdaaeaacaaI WaGaaGimaiaaicdacaaIWaaaaaa@4A41@

(6)

Saturn

56900000000000000000000 0000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaaI1a GaaGOnaiaaiMdacaaIWaGaaGimaiaaicdacaaIWaGaaGimaiaaicda caaIWaGaaGimaiaaicdacaaIWaGaaGimaiaaicdacaaIWaGaaGimai aaicdacaaIWaGaaGimaiaaicdacaaIWaGaaGimaaqaaiaaicdacaaI WaGaaGimaiaaicdaaaaa@4991@

(7)

Uranus

86900000000000000000000000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaaiA dacaaI5aGaaGimaiaaicdacaaIWaGaaGimaiaaicdacaaIWaGaaGim aiaaicdacaaIWaGaaGimaiaaicdacaaIWaGaaGimaiaaicdacaaIWa GaaGimaiaaicdacaaIWaGaaGimaiaaicdacaaIWaGaaGimaiaaicda aaa@48D3@

(8)

Neptune

10200000000000000000000 0000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaaIXa GaaGimaiaaikdacaaIWaGaaGimaiaaicdacaaIWaGaaGimaiaaicda caaIWaGaaGimaiaaicdacaaIWaGaaGimaiaaicdacaaIWaGaaGimai aaicdacaaIWaGaaGimaiaaicdacaaIWaGaaGimaaqaaiaaicdacaaI WaGaaGimaiaaicdaaaaa@4980@

(9)

Pluto

13100000000000000000000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaio dacaaIXaGaaGimaiaaicdacaaIWaGaaGimaiaaicdacaaIWaGaaGim aiaaicdacaaIWaGaaGimaiaaicdacaaIWaGaaGimaiaaicdacaaIWa GaaGimaiaaicdacaaIWaGaaGimaiaaicdaaaa@4693@

Solution

We have studied that, a number is written in standard form as a × 10 k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey41aqRaaG PaVlaaykW7caaIXaGaaGimamaaCaaaleqabaGaam4Aaaaaaaa@3D98@ , where a is terminating decimal and k is an integer.

Sl.No.

Name of the Planet

Mass (in kg)