RD SHARMA Solutions for Class 9 Maths Chapter 2 - Exponents of Real Numbers

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Chapter 2 - Exponents of Real Numbers Ex. 2.1

Solution 1(i)

Solution 1(ii)

Solution 1(iii)

Solution 1(iv)

Solution 1(v)

Solution 1(vi)

Solution 2(i)

Solution 2(ii)

 

Solution 2(iii)

Solution 3(i)

Solution 3(ii)

Solution 4(i)

Solution 4(ii)

Solution 5(i)

Solution 5(ii)

Solution 6

Solution 7(i)

Solution 7(ii)

Solution 7(iii)

Solution 7(iv)

Solution 8(i)

Solution 8(ii)

Solution 8(iii)

Solution 8(iv)

Solution 8(v)

Solution 8(vi)

Solution 9(i)

Solution 9(ii)

Solution 10

Solution 11

Solution 12

Solution 13

Chapter 2 - Exponents of Real Numbers Ex. 2.2

Solution 1(i)

Solution 1(ii)

Solution 1(iii)

Solution 1(iv)

Solution 1(v)

Solution 1(vi)

Solution 1(vii)

Solution 2(i)

Solution 2(ii)

Solution 2(iii)

Solution 2(iv)

Solution 2(v)

Solution 2(vi)

Solution 2(vii)

Solution 3(i)

Solution 3(ii)

Solution 3(iii)

Solution 3(iv)

Solution 3(v)

Solution 3(vi)

Solution 3(vii)

Solution 3(viii)

Solution 3(ix)

Solution 4(i)

Solution 4(ii)

Solution 4(iii)

Solution 4(iv)

 

Note: Question modified

Solution 4(v)

Solution 4(vi)

Solution 4(vii)

Solution 4(viii)

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10(i)

Solution 10(ii)

Solution 10(iii)

Solution 10(iv)

Solution 10(v)

Solution 10(vi)

Solution 10(vii)

Solution 10(viii)

Solution 10(ix)

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16(i)

Solution 16(ii)

Solution 16(iii)

Solution 16(iv)

Solution 16(v)

Solution 16(vi)

Solution 17

Solution 18(i)

Solution 18(ii)

Solution 19

Solution 20

Solution 21(i)

Solution 21(ii)

Solution 22

Solution 23(i)

Solution 23(ii)

Chapter 2 - Exponents of Real Numbers 2.29

Solution 1

{2 - 3(2 - 3)3}3 

= {2 - 3(-1)3}3

= {2 - 3(-1)}3

= {2 - (-3)}3 

= {2 + 3}3

= {5)3 

= 53 

= 125

 

So, correct option is (b).


Solution 2

x = 2, y = -2

x - y = 2 - (-2) = 2 + 2 = 4

Now x - yx-y = 2 - (-2)4 = 2 - 16 = -14

So, correct option is (d).

Solution 3

Square space root space of space straight x space equals space square root of straight x equals straight x to the power of 1 half end exponent space Cube space root space of space straight x space equals cube root of straight x equals space straight x to the power of 1 third end exponent space Thus comma straight x to the power of 1 half end exponent space cross times space straight x to the power of 1 third end exponent equals space left parenthesis straight x right parenthesis to the power of 1 half plus 1 third end exponent space equals space straight x to the power of 5 over 6 end exponent equals space left parenthesis straight x to the power of 5 right parenthesis to the power of 1 over 6 end exponent equals root index 6 of straight x to the power of 5 end root equals sixth space root space of space the space fifth space power space of space straight x Hence comma space correct space option space is space left parenthesis straight b right parenthesis.

Solution 4

Seventh space root space of space straight x space equals space straight x to the power of 1 over 7 end exponent Eighth space root space of space straight x space equals space straight x to the power of 1 over 8 end exponent Now comma space straight x to the power of 1 over 7 end exponent over straight x to the power of 1 over 8 end exponent equals open parentheses straight x close parentheses to the power of 1 over 7 minus 1 over 8 end exponent equals space straight x to the power of 1 over 56 end exponent equals space root index 56 of straight x Hence comma space correct space option space is space left parenthesis straight c right parenthesis.

Solution 5

Square space root space of space 64 space equals space square root of 64 equals space 8 Cube space root space of space 64 space equals space cube root of 64 space equals space 4 Now space fraction numerator square root of 64 over denominator cube root of 64 end fraction equals 8 over 4 equals 2 Hence comma space correct space option space is space left parenthesis straight b right parenthesis.

Chapter 2 - Exponents of Real Numbers 2.30

Solution 6

open curly brackets open parentheses 5 over 6 close parentheses to the power of 1 divided by 5 end exponent close curly brackets to the power of negative 1 divided by 6 end exponent equals 1 over open curly brackets open parentheses begin display style 5 over 6 end style close parentheses to the power of 1 divided by 5 end exponent close curly brackets to the power of 1 divided by 6 end exponent space space space space space left parenthesis option space straight b right parenthesis open curly brackets open parentheses 5 over 6 close parentheses to the power of 1 divided by 5 end exponent close curly brackets to the power of negative 1 divided by 6 end exponent equals 1 over open curly brackets open parentheses begin display style 5 over 6 end style close parentheses to the power of 1 divided by 5 end exponent close curly brackets to the power of 1 divided by 6 end exponent equals 1 over open parentheses begin display style 5 over 6 end style close parentheses to the power of begin display style 1 fifth cross times 1 over 6 end style end exponent equals 1 over open parentheses begin display style 5 over 6 end style close parentheses to the power of begin display style 1 over 30 end style end exponent space space space space space left parenthesis option space straight c right parenthesis open parentheses 5 over 6 close parentheses to the power of 1 fifth cross times open parentheses fraction numerator negative 1 over denominator 6 end fraction close parentheses end exponent space equals space open parentheses 5 over 6 close parentheses to the power of fraction numerator negative 1 over denominator 30 end fraction end exponent space space space space space space space left parenthesis o p t i o n space d right parenthesis But comma space open curly brackets open parentheses 5 over 6 close parentheses to the power of 1 divided by 5 end exponent close curly brackets to the power of negative 1 divided by 6 end exponent space not equal to space open parentheses 5 over 6 close parentheses to the power of 1 fifth minus 1 over 6 end exponent space space space space space left parenthesis option space straight a right parenthesis Hence comma space correct space option space is space left parenthesis straight a right parenthesis.

Solution 7

open parentheses straight x to the power of negative 1 end exponent plus straight y to the power of negative 1 end exponent close parentheses to the power of negative 1 end exponent equals space open parentheses 1 over straight x plus 1 over straight y close parentheses to the power of negative 1 end exponent equals open parentheses fraction numerator straight x plus straight y over denominator xy end fraction close parentheses to the power of negative 1 end exponent space space equals fraction numerator xy over denominator straight x plus straight y end fraction Hence comma space correct space option space is space left parenthesis straight c right parenthesis.

Solution 8

8x+1 = 64 = (8)2

so, x + 1 = 2

Hence, x = 1

Now, 32x + 1 = 32(1) + 1  = 33 = 27

Hence, correct option is (d).

Solution 9

left parenthesis 2 cubed right parenthesis squared equals left parenthesis 2 squared right parenthesis cubed space space space space space space space space space space space open square brackets because open parentheses straight a to the power of straight b close parentheses to the power of straight c space equals space open parentheses straight a to the power of straight b close parentheses to the power of straight c space space by space properties close square brackets therefore space left parenthesis 2 cubed right parenthesis squared space space equals space 4 cubed space equals 4 to the power of straight x rightwards double arrow straight x space equals space 3 Now space 3 to the power of straight x space equals space 3 cubed space equals 27 Hence comma space correct space option space is space left parenthesis straight d right parenthesis.

Solution 10

straight x to the power of negative 2 end exponent space equals space 64 rightwards double arrow straight x squared space equals space 1 over 64 space rightwards double arrow straight x space equals space 1 over 8 Now comma space straight x to the power of 1 divided by 3 end exponent plus straight x to the power of 0 space space equals space open parentheses 1 over 8 close parentheses to the power of 1 divided by 3 end exponent plus open parentheses 1 over 8 close parentheses to the power of 0 equals open parentheses 1 over 8 close parentheses to the power of 1 divided by 3 end exponent plus 1 equals open curly brackets open parentheses 1 half close parentheses to the power of up diagonal strike 3 end exponent close curly brackets to the power of 1 divided by up diagonal strike 3 end exponent plus 1 equals 1 half plus 1 equals 3 over 2 Hence comma space correct space option space is space left parenthesis straight c right parenthesis.

Solution 11

open parentheses fraction numerator negative 1 over denominator 27 end fraction close parentheses to the power of negative 2 divided by 3 end exponent equals space open parentheses negative 27 close parentheses to the power of 2 divided by 3 end exponent equals open square brackets open parentheses negative 3 close parentheses cubed close square brackets to the power of bevelled 2 over 3 end exponent equals space open parentheses negative 3 close parentheses squared equals space 9 Hence comma space correct space option space is space left parenthesis straight a right parenthesis.

Solution 12

Option space left parenthesis straight a right parenthesis colon cube root of 2 not equal to space cube root of 8 rightwards double arrow open parentheses cube root of 2 close parentheses to the power of negative 1 divided by 2 end exponent not equal to open parentheses cube root of 8 close parentheses to the power of negative 1 divided by 2 end exponent Option space left parenthesis straight b right parenthesis colon cube root of 8 equals open parentheses 8 close parentheses to the power of 1 divided by 3 end exponent rightwards double arrow open parentheses cube root of 8 close parentheses to the power of negative 1 divided by 2 end exponent equals 8 to the power of negative 1 divided by 6 end exponent Option space left parenthesis straight c right parenthesis colon open parentheses cube root of 8 close parentheses to the power of negative 1 divided by 2 end exponent space equals 1 over open parentheses cube root of 8 close parentheses to the power of 1 divided by 2 end exponent Option space left parenthesis straight d right parenthesis colon cube root of 8 equals 8 to the power of 1 divided by 3 end exponent space equals space open parentheses 2 to the power of up diagonal strike 3 end exponent close parentheses to the power of 1 divided by up diagonal strike 3 end exponent equals 2 rightwards double arrow open parentheses cube root of 8 close parentheses to the power of negative 1 divided by 2 end exponent equals open parentheses 2 close parentheses to the power of negative 1 divided by 2 end exponent space space equals space 1 over open parentheses 2 close parentheses to the power of 1 divided by 2 end exponent equals fraction numerator 1 over denominator square root of 2 end fraction Hence comma space correct space option space is space left parenthesis straight a right parenthesis.

Solution 13

Option space left parenthesis straight a right parenthesis colon open parentheses 100 over 9 close parentheses to the power of negative 3 divided by 2 end exponent equals 1 over open parentheses 100 over 9 close parentheses to the power of 3 divided by 2 end exponent equals open parentheses 9 over 100 close parentheses to the power of 3 divided by 2 end exponent space space space space space space open curly brackets because space straight a to the power of negative straight x end exponent equals 1 over straight a to the power of straight x close curly brackets space Option space left parenthesis straight b right parenthesis colon space open parentheses 100 over 9 close parentheses to the power of negative 3 divided by 2 end exponent equals 1 over open parentheses 100 over 9 close parentheses to the power of 3 divided by 2 end exponent space space space space space open curly brackets because space straight a to the power of negative straight x end exponent equals 1 over straight a to the power of straight x close curly brackets Option space left parenthesis straight c right parenthesis colon open parentheses 100 over 9 close parentheses to the power of negative 3 divided by 2 end exponent equals open parentheses 9 over 100 close parentheses to the power of 3 divided by 2 end exponent equals open curly brackets open parentheses 3 over 10 close parentheses to the power of up diagonal strike 2 end exponent close curly brackets to the power of 3 divided by up diagonal strike 2 end exponent equals open parentheses 3 over 10 close parentheses cubed equals 3 over 10 cross times 3 over 10 cross times 3 over 10 space space Option space left parenthesis straight d right parenthesis colon open parentheses 100 over 9 close parentheses to the power of negative 3 divided by 2 end exponent equals open parentheses 9 over 100 close parentheses to the power of 3 divided by 2 end exponent equals space open curly brackets open parentheses 9 over 100 close parentheses cubed close curly brackets to the power of 1 divided by 2 end exponent equals space square root of 9 over 100 cross times 9 over 100 cross times 9 over 100 end root space not equal to square root of 100 over 9 cross times 100 over 9 cross times 100 over 9 end root Hence comma space correct space option space is space left parenthesis straight d right parenthesis.

Solution 14

straight a to the power of negative 1 end exponent equals 1 over straight a comma space space straight b to the power of negative 1 end exponent equals 1 over straight b comma space space straight c to the power of negative 1 end exponent equals 1 over straight c rightwards double arrow square root of straight a to the power of negative 1 end exponent straight b end root equals square root of straight b over straight a end root comma space space square root of straight b to the power of negative 1 end exponent straight c end root equals square root of straight c over straight b end root comma space space square root of straight c to the power of negative 1 end exponent straight a end root equals square root of straight a over straight c end root Now comma square root of straight a to the power of negative 1 end exponent straight b end root space cross times space square root of straight b to the power of negative 1 end exponent straight c end root space cross times space square root of straight c to the power of negative 1 end exponent straight a end root space equals space square root of straight b over straight a end root cross times square root of straight c over straight b end root cross times square root of straight a over straight c end root equals fraction numerator square root of straight b over denominator square root of straight a end fraction cross times fraction numerator square root of straight c over denominator square root of straight b end fraction cross times fraction numerator square root of straight a over denominator square root of straight c end fraction equals 1 Hence comma space correct space option space is space left parenthesis straight a right parenthesis.

Solution 15

open parentheses 2 over 3 close parentheses to the power of straight x equals open parentheses 3 over 2 close parentheses to the power of negative straight x end exponent space space space space space space space space space space open curly brackets because space straight a to the power of straight x equals 1 over straight a to the power of negative straight x end exponent close curly brackets therefore open parentheses 2 over 3 close parentheses to the power of straight x space cross times space open parentheses 3 over 2 close parentheses to the power of 2 straight x end exponent equals open parentheses 3 over 2 close parentheses to the power of negative straight x end exponent space space open parentheses 3 over 2 close parentheses to the power of 2 straight x end exponent equals open parentheses 3 over 2 close parentheses to the power of 2 straight x minus straight x end exponent space equals open parentheses 3 over 2 close parentheses to the power of straight x Thus comma open parentheses 3 over 2 close parentheses to the power of straight x equals 81 over 16 rightwards double arrow open parentheses 3 over 2 close parentheses to the power of straight x space equals 3 to the power of 4 over 2 to the power of 4 equals open parentheses 3 over 2 close parentheses to the power of 4 rightwards double arrow straight x equals 4 Hence comma space correct space option space is space left parenthesis straight c right parenthesis.

Chapter 2 - Exponents of Real Numbers 2.31

Solution 16

8 to the power of negative 4 divided by 3 end exponent space equals space open parentheses 2 to the power of up diagonal strike 3 end exponent close parentheses to the power of negative 4 divided by up diagonal strike 3 end exponent space equals space 2 to the power of negative 4 end exponent So comma space 8 to the power of negative 4 divided by 3 end exponent space divided by space 2 to the power of negative 2 end exponent space equals space 2 to the power of negative 4 end exponent over 2 to the power of negative 2 end exponent equals 2 to the power of negative 2 end exponent Now comma open curly brackets 8 to the power of negative 4 divided by 3 end exponent space divided by space 2 to the power of negative 2 end exponent close curly brackets to the power of 1 divided by 2 end exponent space equals space open curly brackets 2 to the power of negative up diagonal strike 2 end exponent close curly brackets to the power of 1 divided by up diagonal strike 2 end exponent space equals 2 to the power of negative 1 end exponent equals 1 half Hence comma space correct space option space is space left parenthesis straight a right parenthesis.

Solution 17

3125 space straight a to the power of 10 straight b to the power of 5 straight c to the power of 10 space equals space 5 to the power of 5 left parenthesis straight a squared right parenthesis to the power of 5 space left parenthesis straight b right parenthesis to the power of 5 space left parenthesis straight c squared right parenthesis to the power of 5 Now comma space fifth root of 3125 space straight a to the power of 10 space straight b to the power of 5 space straight c to the power of 10 end root space equals space left parenthesis 3125 cross times straight a to the power of 10 cross times straight b to the power of 5 cross times straight c to the power of 10 right parenthesis to the power of 1 divided by 5 end exponent equals open square brackets 5 to the power of 5 cross times open parentheses straight a squared close parentheses to the power of 5 cross times open parentheses straight b close parentheses to the power of 5 cross times open parentheses straight c squared close parentheses to the power of 5 close square brackets to the power of 1 divided by 5 end exponent equals open square brackets open parentheses 5 straight a squared bc squared close parentheses to the power of 5 close square brackets to the power of 1 divided by 5 end exponent equals 5 straight a squared bc squared Hence comma space correct space option space is space left parenthesis straight a right parenthesis. space space space space space space space space space

Solution 18

open curly brackets straight m-th root of straight n-th root of straight a end root close curly brackets to the power of mn equals open curly brackets straight m-th root of open parentheses straight a close parentheses to the power of 1 divided by straight n end exponent end root close curly brackets to the power of mn equals open curly brackets open square brackets open parentheses straight a close parentheses to the power of 1 divided by straight n end exponent close square brackets to the power of 1 divided by straight m end exponent close curly brackets to the power of mn equals open curly brackets open parentheses straight a close parentheses to the power of 1 divided by mn end exponent close curly brackets to the power of mn equals straight a Hence comma space correct space option space is space left parenthesis straight b right parenthesis.

Solution 19

straight x equals 2 space and space straight y equals 4 Now comma open parentheses straight x over straight y close parentheses to the power of straight x minus straight y end exponent plus open parentheses straight y over straight x close parentheses to the power of straight y minus straight x end exponent equals open parentheses 2 over 4 close parentheses to the power of 2 minus 4 end exponent plus open parentheses 4 over 2 close parentheses to the power of 4 minus 2 end exponent equals open parentheses 1 half close parentheses to the power of negative 2 end exponent plus open parentheses 2 close parentheses squared equals open parentheses 2 close parentheses squared plus open parentheses 2 close parentheses squared equals 4 plus 4 equals 8 Hence comma space correct space option space is space left parenthesis straight b right parenthesis.

Solution 20

open curly brackets open square brackets open parentheses 1 over 7 squared close parentheses to the power of negative 2 end exponent close square brackets to the power of negative 1 divided by 3 end exponent close curly brackets to the power of 1 divided by 4 end exponent equals 7 to the power of straight m rightwards double arrow open curly brackets open square brackets open parentheses 7 to the power of negative 2 end exponent close parentheses to the power of negative 2 end exponent close square brackets to the power of negative 1 divided by 3 end exponent close curly brackets to the power of 1 divided by 4 end exponent equals 7 to the power of straight m rightwards double arrow open curly brackets open square brackets open parentheses 7 close parentheses to the power of 4 close square brackets to the power of negative 1 divided by 3 end exponent close curly brackets to the power of 1 divided by 4 end exponent equals 7 to the power of straight m rightwards double arrow open curly brackets open parentheses 7 close parentheses to the power of negative 4 divided by 3 end exponent close curly brackets to the power of 1 divided by 4 end exponent equals 7 to the power of straight m rightwards double arrow open curly brackets open parentheses 7 close parentheses to the power of negative 4 divided by 3 end exponent close curly brackets to the power of 1 divided by 4 equals end exponent equals 7 to the power of straight m rightwards double arrow open curly brackets open parentheses 7 close parentheses to the power of negative 1 divided by 3 end exponent close curly brackets equals 7 to the power of straight m rightwards double arrow negative 1 third equals straight m Hence comma space correct space option space is space left parenthesis straight a right parenthesis.

Solution 21

open curly brackets open parentheses 23 plus 2 squared close parentheses to the power of 2 divided by 3 end exponent plus open parentheses 140 minus 19 close parentheses to the power of 1 divided by 2 end exponent close curly brackets squared equals open curly brackets open parentheses 23 plus 4 close parentheses to the power of 2 divided by 3 end exponent plus open parentheses 121 close parentheses to the power of 1 divided by 2 end exponent close curly brackets squared equals open curly brackets open parentheses 27 close parentheses to the power of 2 divided by 3 end exponent plus open parentheses 121 close parentheses to the power of 1 divided by 2 end exponent close curly brackets squared equals open curly brackets open parentheses 3 cubed close parentheses to the power of 2 divided by 3 end exponent plus open parentheses 11 squared close parentheses to the power of 1 divided by 2 end exponent close curly brackets squared equals open curly brackets 3 squared plus 11 close curly brackets squared equals open curly brackets 9 plus 11 close curly brackets squared equals open curly brackets 20 close curly brackets squared equals 400 Hence comma space correct space option space is space left parenthesis straight d right parenthesis.

Solution 22

straight a to the power of straight p cross times straight a to the power of straight n space equals space straight a to the power of straight p plus straight n end exponent therefore space open parentheses 256 close parentheses to the power of 0.16 end exponent cross times open parentheses 256 close parentheses to the power of 0.09 end exponent space space space space space equals open parentheses 256 close parentheses to the power of 0.16 plus 0.09 end exponent space space space space space equals open parentheses 256 close parentheses to the power of 0.25 end exponent space space space space space space equals open parentheses 256 close parentheses to the power of 1 fourth end exponent space space space space space space space space space space space space equals open parentheses 4 to the power of 4 close parentheses to the power of 1 fourth end exponent space space space space space equals 4 So comma space correct space option space is space left parenthesis straight a right parenthesis.

Solution 23

10 to the power of 2 straight y end exponent space equals space open parentheses 10 squared close parentheses to the power of straight y equals 100 to the power of straight y equals 10 to the power of straight y.10 to the power of straight y equals open parentheses 10 to the power of straight y close parentheses squared But comma space 10 to the power of 2 straight y end exponent equals 25 rightwards double arrow open parentheses 10 to the power of straight y close parentheses squared equals 25 rightwards double arrow open parentheses 10 to the power of straight y close parentheses squared equals open parentheses 5 close parentheses squared rightwards double arrow 10 to the power of straight y equals 5 rightwards double arrow fraction numerator 1 over denominator 10 straight y end fraction equals 1 fifth rightwards double arrow 10 to the power of negative straight y end exponent equals 1 fifth So comma space correct space option space is space left parenthesis straight d right parenthesis.

Solution 24

9 to the power of straight x plus 2 end exponent space equals space 240 plus 9 to the power of straight x rightwards double arrow 9 to the power of straight x space cross times space 9 squared space equals space 240 plus 9 to the power of straight x space space rightwards double arrow 9 to the power of straight x space cross times space 81 equals 240 plus 9 to the power of straight x space space rightwards double arrow 9 to the power of straight x left parenthesis 81 right parenthesis space minus space 9 to the power of straight x equals 240 rightwards double arrow 9 to the power of straight x left parenthesis 81 space minus space 1 right parenthesis equals 240 space space rightwards double arrow 9 to the power of straight x space cross times space 80 space equals space 240 rightwards double arrow 9 to the power of straight x equals 240 over 80 equals 3 equals left parenthesis 3 squared right parenthesis to the power of 1 half end exponent equals left parenthesis 9 right parenthesis to the power of 1 half end exponent rightwards double arrow straight x equals 1 half equals 0.5 Hence comma space correct space option space is space left parenthesis straight a right parenthesis.

Solution 25

straight x squared space equals space 2 therefore space straight x space equals space square root of 2 Now comma straight x cubed space equals space straight x squared space. space straight x space equals space 2 square root of 2 Hence comma space correct space option space is space left parenthesis straight b right parenthesis. space space space space space space

Solution 26

straight x over straight x to the power of 1.5 end exponent equals 8 straight x to the power of negative 1 end exponent rightwards double arrow open parentheses straight x close parentheses to the power of 1 minus 1.5 end exponent space equals 8 over straight x rightwards double arrow straight x to the power of negative 0.5 end exponent equals 8 over straight x rightwards double arrow straight x to the power of negative 1 divided by 2 end exponent equals 8 over straight x rightwards double arrow straight x. straight x to the power of negative 1 divided by 2 end exponent equals 8 rightwards double arrow straight x to the power of 1 minus 1 half end exponent equals 8 rightwards double arrow space straight x to the power of 1 divided by 2 end exponent equals 8 rightwards double arrow open parentheses straight x to the power of 1 divided by 2 end exponent close parentheses squared space space equals open parentheses 8 close parentheses squared rightwards double arrow straight x space equals space 64 Hence comma space correct space option space is space left parenthesis straight d right parenthesis.

Chapter 2 - Exponents of Real Numbers 2.32

Solution 28

4 to the power of straight x space minus space 4 to the power of straight x minus 1 end exponent space equals space 24 rightwards double arrow 4 to the power of straight x minus 1 end exponent open parentheses 4 minus 1 close parentheses equals 24 rightwards double arrow 4 to the power of straight x minus 1 end exponent space open parentheses 3 close parentheses equals 24 rightwards double arrow 4 to the power of straight x minus 1 end exponent space equals space 8 rightwards double arrow 4 to the power of straight x minus 1 end exponent space equals 8 rightwards double arrow 2 to the power of 2 left parenthesis straight x minus 1 right parenthesis end exponent equals space 2 cubed space space space rightwards double arrow 2 left parenthesis straight x minus 1 right parenthesis equals 3 rightwards double arrow straight x equals 3 over 2 plus 1 rightwards double arrow straight x equals 5 over 2 Now comma space open parentheses 2 straight x close parentheses to the power of straight x space end exponent equals open parentheses up diagonal strike 2 cross times fraction numerator 5 over denominator up diagonal strike 2 end fraction close parentheses to the power of 5 divided by 2 end exponent equals 5 to the power of 5 divided by 2 end exponent equals open parentheses 5 to the power of 5 close parentheses to the power of 1 divided by 2 end exponent equals square root of 5 to the power of 5 end root equals space space 5 squared square root of 5 space equals 25 square root of 5 Hence comma space correct space option space is space left parenthesis straight c right parenthesis.

Solution 27

straight g space equals space straight t to the power of 2 divided by 3 end exponent space plus space 4 straight t to the power of negative 1 divided by 2 end exponent Substituting space straight t space equals space 64 comma space we space get straight g equals left parenthesis 64 right parenthesis to the power of 2 divided by 3 end exponent space plus space 4 left parenthesis 64 right parenthesis to the power of negative 1 divided by 2 end exponent space space equals left parenthesis 4 cubed right parenthesis to the power of 2 divided by 3 end exponent space plus space 4 left parenthesis 8 squared right parenthesis to the power of negative 1 divided by 2 end exponent space space equals space 4 squared space plus space 4 left parenthesis 8 right parenthesis to the power of negative 1 end exponent space space equals 16 plus 4 over 8 space space equals 16 plus 1 half space equals 33 over 2 Hence comma space correct space option space is space left parenthesis straight b right parenthesis.

Solution 29

open parentheses 256 close parentheses to the power of negative open parentheses 4 to the power of negative 3 divided by 2 end exponent close parentheses end exponent equals space open parentheses 2 to the power of 8 close parentheses to the power of negative open parentheses 2 squared close parentheses to the power of negative 3 divided by 2 end exponent end exponent equals space open parentheses 2 to the power of 8 close parentheses to the power of negative open parentheses 2 close parentheses to the power of negative 3 end exponent end exponent equals space open parentheses 2 to the power of 8 close parentheses to the power of negative 1 divided by 8 end exponent equals 2 to the power of negative 1 end exponent equals 1 half Hence comma space correct space option space is space left parenthesis straight d right parenthesis.

Solution 30

3 to the power of 2 straight x minus 8 end exponent over 225 equals 5 cubed over 5 to the power of straight x rightwards double arrow fraction numerator 3 to the power of 2 straight x minus 8 end exponent over denominator 25 cross times 9 end fraction equals 5 cubed over 5 to the power of straight x rightwards double arrow fraction numerator 3 to the power of 2 straight x minus 8 end exponent over denominator 5 squared cross times 3 squared end fraction equals 5 cubed over 5 to the power of straight x rightwards double arrow 3 to the power of 2 straight x minus 8 end exponent space equals space fraction numerator 3 squared.5 squared.5 cubed over denominator 5 to the power of straight x end fraction rightwards double arrow 3 to the power of 2 straight x minus 8 end exponent space equals open parentheses 3 squared close parentheses open parentheses 5 to the power of 5 minus straight x end exponent close parentheses rightwards double arrow 5 to the power of 5 minus straight x end exponent equals 3 to the power of 2 straight x minus 8 end exponent over 3 squared rightwards double arrow 1 over 5 to the power of open parentheses straight x minus 5 close parentheses end exponent equals 3 to the power of 2 straight x minus 10 end exponent space rightwards double arrow 1 over 5 to the power of open parentheses straight x minus 5 close parentheses end exponent equals open parentheses 3 squared close parentheses to the power of open parentheses straight x minus 5 close parentheses end exponent rightwards double arrow 1 over 5 to the power of open parentheses straight x minus 5 close parentheses end exponent equals open parentheses 9 close parentheses to the power of straight x minus 5 end exponent rightwards double arrow 1 equals open parentheses 5 cross times 9 close parentheses to the power of straight x minus 5 end exponent space space space space space space space space space space space space rightwards double arrow straight x minus 5 space equals space 0 space space space space space space space space.... open parentheses If space straight a to the power of straight p equals 1 comma space then space straight p space equals space 0 close parentheses rightwards double arrow straight x equals 5 Hence comma space correct space option space is space left parenthesis straight c right parenthesis.

Solution 31

Correct space option space colon space left parenthesis straight c right parenthesis 64 to the power of negative 1 divided by 3 end exponent space open parentheses 64 to the power of 1 divided by 3 end exponent space minus space 64 to the power of 2 divided by 3 end exponent close parentheses space equals space open parentheses 4 to the power of up diagonal strike 3 end exponent close parentheses to the power of negative 1 divided by up diagonal strike 3 end exponent space space open curly brackets open parentheses 4 up diagonal strike blank cubed end strike close parentheses to the power of 1 divided by up diagonal strike 3 end exponent space minus open parentheses 4 up diagonal strike blank cubed end strike close parentheses to the power of 2 divided by up diagonal strike 3 end exponent space space close curly brackets equals 4 to the power of negative 1 end exponent open parentheses 4 to the power of 1 space minus space 4 squared close parentheses equals space 1 fourth open parentheses 4 minus 16 close parentheses equals fraction numerator negative 12 over denominator 4 end fraction equals negative 3 Hence comma space correct space option space is space left parenthesis straight c right parenthesis. space

Solution 32

square root of 5 to the power of straight n end root equals 125 rightwards double arrow open parentheses 5 to the power of straight n close parentheses to the power of 1 divided by 2 end exponent space equals space 5 cubed rightwards double arrow 5 to the power of straight n divided by 2 end exponent equals space 5 cubed rightwards double arrow straight n over 2 equals 3 rightwards double arrow straight n equals 6 Now comma 5 to the power of straight n-th root of 64 end exponent equals open parentheses 5 close parentheses to the power of left parenthesis 64 right parenthesis to the power of 1 divided by straight n end exponent end exponent equals left parenthesis 5 right parenthesis to the power of left parenthesis 64 right parenthesis to the power of 1 divided by 6 end exponent end exponent equals left parenthesis 5 right parenthesis to the power of open parentheses 2 to the power of 6 close parentheses to the power of 1 divided by 6 end exponent end exponent equals 5 squared equals 25 Hence comma space correct space option space is space left parenthesis straight a right parenthesis.

Solution 33

open parentheses 16 close parentheses to the power of 2 straight x plus 3 end exponent equals open parentheses 64 close parentheses to the power of straight x plus 3 end exponent rightwards double arrow open parentheses 4 squared close parentheses to the power of 2 straight x plus 3 end exponent equals space open parentheses 4 cubed close parentheses to the power of straight x plus 3 end exponent rightwards double arrow open parentheses 4 close parentheses to the power of 2 open parentheses 2 straight x plus 3 close parentheses end exponent equals open parentheses 4 to the power of 3 open parentheses straight x plus 3 close parentheses end exponent close parentheses space space space rightwards double arrow 2 open parentheses 2 straight x plus 3 close parentheses equals 3 open parentheses straight x plus 3 close parentheses rightwards double arrow 4 straight x plus 6 equals 3 straight x plus 9 space rightwards double arrow straight x equals 3 Now comma space 4 to the power of 2 straight x minus 2 end exponent equals 4 to the power of 2 cross times 3 minus 2 end exponent equals 4 to the power of 4 equals 256 Hence comma space correct space option space is space left parenthesis straight b right parenthesis.

Solution 34

2 to the power of negative straight m end exponent cross times 1 over 2 to the power of straight m equals 1 fourth rightwards double arrow 2 to the power of negative straight m end exponent cross times 2 to the power of negative straight m end exponent equals 1 over 2 squared rightwards double arrow 2 to the power of negative 2 straight m end exponent space equals 2 to the power of negative 2 end exponent rightwards double arrow negative 2 straight m equals negative 2 rightwards double arrow straight m equals 1 Now comma 1 over 14 open curly brackets open parentheses 4 to the power of straight m close parentheses to the power of 1 divided by 2 end exponent space plus space open parentheses 1 over 5 to the power of straight m close parentheses to the power of negative 1 end exponent close curly brackets equals 1 over 14 open curly brackets open parentheses 4 close parentheses to the power of 1 divided by 2 end exponent space plus space open parentheses 1 fifth close parentheses to the power of negative 1 end exponent close curly brackets equals 1 over 14 open curly brackets open parentheses 2 squared close parentheses to the power of 1 divided by 2 end exponent space plus 5 close curly brackets equals 1 over 14 open parentheses 2 plus 5 close parentheses equals 7 over 14 equals 1 half Hence comma space correct space option space is space left parenthesis straight a right parenthesis.

Solution 35

2 to the power of straight m plus straight n end exponent over 2 to the power of straight n minus straight m end exponent space equals 16 rightwards double arrow 2 to the power of straight m plus straight n minus open parentheses straight n minus straight m close parentheses end exponent space equals 2 to the power of 4 rightwards double arrow 2 to the power of 2 straight m end exponent equals space 2 to the power of 4 rightwards double arrow 2 straight m equals 4 rightwards double arrow straight m equals 2 space space space space space space space space space space space...... left parenthesis 1 right parenthesis And comma space 3 to the power of straight p over 3 to the power of straight n equals 81 rightwards double arrow 3 to the power of straight p minus straight n end exponent equals 3 to the power of 4 rightwards double arrow straight p space minus space straight n equals 4 rightwards double arrow straight n space minus space straight p equals negative 4 space space space..... left parenthesis 2 right parenthesis Now comma space straight a to the power of 2 straight m plus straight n minus straight p end exponent over open parentheses straight a to the power of straight m minus 2 straight n plus 2 straight p end exponent close parentheses to the power of negative 1 end exponent equals straight a to the power of 2 straight m space plus space open parentheses straight n space minus space straight p close parentheses end exponent over open parentheses straight a to the power of straight m minus 2 open parentheses straight n minus straight p close parentheses end exponent close parentheses to the power of negative 1 end exponent equals straight a to the power of 2 open parentheses 2 close parentheses plus open parentheses negative 4 close parentheses end exponent over open parentheses straight a to the power of 2 minus 2 open parentheses negative 4 close parentheses end exponent close parentheses to the power of negative 1 end exponent space space space space space space space space space space space.... left square bracket From space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis right square bracket equals straight a to the power of 4 minus 4 end exponent over open parentheses straight a to the power of 2 plus 8 end exponent close parentheses to the power of negative 1 end exponent equals straight a to the power of 0 over straight a to the power of negative 10 end exponent equals space straight a to the power of 10 equals space open parentheses 2 to the power of 1 divided by 10 end exponent close parentheses to the power of 10 equals 2 Hence comma space correct space option space is space left parenthesis straight a right parenthesis.

Solution 36

fraction numerator 3 to the power of 5 straight x end exponent space cross times 81 squared cross times 6561 over denominator 3 to the power of 2 straight x end exponent end fraction equals 3 to the power of 7 rightwards double arrow fraction numerator 3 to the power of 5 straight x end exponent space over denominator 3 to the power of 2 straight x end exponent end fraction cross times 81 squared cross times 6561 equals 3 to the power of 7 rightwards double arrow 3 to the power of 5 straight x minus 2 straight x end exponent cross times open parentheses 3 to the power of 4 close parentheses squared cross times open parentheses 81 close parentheses squared equals 3 to the power of 7 rightwards double arrow 3 to the power of 3 straight x end exponent cross times 3 to the power of 8 cross times 3 to the power of 8 equals 3 to the power of 7 rightwards double arrow 3 open parentheses blank to the power of 3 straight x plus 8 plus 8 end exponent close parentheses equals 3 to the power of 7 rightwards double arrow 3 straight x plus 8 plus 8 equals 7 rightwards double arrow 3 straight x plus 16 equals 7 rightwards double arrow 3 straight x equals negative 9 rightwards double arrow straight x equals negative 3 Hence comma space correct space option space is space left parenthesis straight b right parenthesis.

Chapter 2 - Exponents of Real Numbers 2.33

Solution 37

Option space left parenthesis straight a right parenthesis space is space incorrect space because space square root of straight x minus square root of straight y not equal to square root of straight x minus straight y end root In space square space root space operation comma space we space can space not space take space square space root space in space common. Option space left parenthesis straight b right parenthesis space is space incorrect space because space square root of straight x plus square root of straight x equals 2 square root of straight x space not equal to square root of 2 straight x end root Option space left parenthesis straight c right parenthesis space is space incorrect space because space in space straight x square root of straight y comma space square root of straight y space is space irrational space and space in space straight y square root of straight x comma space square root of straight x space is space irrational. space Two space numbers space space straight x square root of straight y space and space straight y square root of straight x space will space be space equal space if space and space only space if space straight x equals straight y. But space straight x greater than straight y. Now comma space option space left parenthesis straight d right parenthesis space square root of xy equals square root of straight x. square root of straight y space is space true space because space straight x comma space straight y space greater than space 0 space and space this space is space straight a space properly space of space square space root space operations. Hence comma space correct space option space is space left parenthesis straight d right parenthesis.

Solution 38

10 to the power of straight x space equals space 64 Taking space square space root space on space both space sides comma space we space have open parentheses 10 to the power of straight x close parentheses to the power of 1 divided by 2 end exponent equals space open parentheses 64 close parentheses to the power of 1 divided by 2 end exponent rightwards double arrow 10 to the power of straight x over 2 end exponent equals open parentheses 8 squared close parentheses to the power of 1 divided by 2 end exponent rightwards double arrow 10 to the power of straight x over 2 end exponent equals 8 Multiplying space by space 10 space on space both space sides comma space we space have 10 to the power of straight x over 2 end exponent cross times space 10 space 8 cross times 10 rightwards double arrow 10 to the power of straight x over 2 end exponent cross times space 10 equals 80 rightwards double arrow 10 to the power of straight x over 2 plus 1 end exponent equals space 80 Hence comma space correct space option space is space left parenthesis straight c right parenthesis.

Solution 39

fraction numerator 5 to the power of straight n plus 2 end exponent minus 6 cross times 5 to the power of straight n plus 1 end exponent over denominator 13 cross times 5 to the power of straight n minus 2 cross times 5 to the power of straight n plus 1 end exponent end fraction space space Taking space 5 to the power of straight n space common space from space Numerotor space and space denominator comma space we space have fraction numerator 5 to the power of straight n open parentheses 5 squared minus 6 cross times 5 to the power of 1 close parentheses over denominator 5 to the power of straight n open parentheses 13 minus 2 cross times 5 to the power of 1 close parentheses end fraction equals fraction numerator 25 minus 30 over denominator 13 minus 10 end fraction equals fraction numerator negative 5 over denominator 3 end fraction Hence comma space correct space option space is space left parenthesis straight b right parenthesis.

Solution 40

square root of 2 to the power of straight n end root space equals 1024 rightwards double arrow space open parentheses 2 to the power of straight n close parentheses to the power of 1 divided by 2 end exponent equals 2 to the power of 10 space equals space 2 to the power of straight n divided by 2 end exponent equals space 2 to the power of 10 rightwards double arrow straight n over 2 equals 10 rightwards double arrow space straight n space equals space 20 Now comma space 3 to the power of 2 open parentheses straight n over 2 minus 4 close parentheses end exponent equals 3 to the power of 2 open parentheses 20 over 4 minus 4 close parentheses end exponent equals 3 to the power of 2 open parentheses 5 minus 4 close parentheses end exponent equals 3 squared equals 9 Hence comma space correct space option space is space left parenthesis straight b right parenthesis.