Chapter 2 : Exponents of Real Numbers - Rd Sharma Solutions for Class 9 Maths CBSE

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Chapter 2 - Exponents of Real Numbers Excercise Ex. 2.1

Question 1

Simplify:

3(a4b3)10 × 5(a2b2)3

Solution 1

Question 2

Simplify:

(2x-2y3)3

Solution 2

Question 3

Simplify:

Solution 3

Question 4

Simplify:

Solution 4

Question 5

Simplify:

Solution 5

Question 6

Simplify:

Solution 6

Question 7

If a = 3 and b = -2, find the value of:

aa + bb

Solution 7

Question 8

If a = 3 and b = -2, find the value of:

ab + ba

Solution 8

  

Question 9

If a = 3 and b = -2, find the value of:

(a + b)ab

Solution 9

Question 10

Prove that:

Solution 10

Question 11

Prove that:

Solution 11

Question 12

Prove that:

Solution 12

Question 13

Prove that:

  

Solution 13

Question 14

Prove that:

Solution 14

Question 15

Prove that:

Solution 15

Question 16

Solution 16

Question 17

Simplify:

Solution 17

Question 18

Simplify:

  

Solution 18

Question 19

Simplify:

Solution 19

Question 20

Simplify:

Solution 20

Question 21

Solve the equation for x:

72x + 3 = 1

Solution 21

Question 22

Solve the equation for x:

2x+1 = 4x-3

Solution 22

Question 23

Solve the equation for x:

25x + 3 = 8x + 3

Solution 23

Question 24

Solve the equation for x:

Solution 24

Question 25

Solve the equation for x:

Solution 25

Question 26

Solve the equation for x:

23x - 7 = 256

Solution 26

Question 27

Solve the equation for x:

22x - 2x+3 + 24 = 0

Solution 27

Question 28

Solve the equation for x:

32x + 4 + 1 = 2.3x + 2

Solution 28

Question 29

If 49392 = a4b2c3, find the values of a, b and c, where a, b and c are different positive primes.

Solution 29

Question 30

If 1176 = 2a × 3b × 7c, find a, b and c.

Solution 30

Question 31

Given 4725 = 3a 5b 7c , find

  1. the integral values of a, b and c
  2. the values of 2-a3b7c
Solution 31

Question 32

If a = xyp - 1, b = xyq-1 and c = xyr-1, prove that aq-rbr-p cp-q = 1.

Solution 32

Chapter 2 - Exponents of Real Numbers Excercise Ex. 2.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Assuming that x, y, z are positive real numbers, simplify each of the following:

Solution 7

Question 8

Solution 8

Question 9

Simplify:

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Prove that:

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Show that:

Solution 24

Question 25

Show that:

Solution 25

Question 26

Show that:

Solution 26

Question 27

Show that:

Note: Question modified

Solution 27

 

Note: Question modified

Question 28

Show that:

(xa-b)a+b(xb-c)b+c(xc-a)c+a = 1

Solution 28

Question 29

Show that:

Solution 29

Question 30

Show that:

Solution 30

Question 31

Show that:

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Find the value of x if:

Solution 42

Question 43

Find the value of x if:

52x + 3 = 1

Solution 43

Question 44

Find the value of x if:

Solution 44

Question 45

Find the value of x if:

Solution 45

Question 46

If x = 21/3 + 22/3, show that x3 - 6x = 6.

Solution 46

Question 47

Determine (8x)x, if 9x+2 = 240 + 9x.

Solution 47

Question 48

If 3x+1 = 9x-2, find the value of 21+x.

Solution 48

Question 49

If 34x = (81)-1 and 101/y = 0.0001, find the value of  2-x+4y.

Solution 49

Question 50

If 53x = 125 and 10y = 0.001 find x and y.

Solution 50

Question 51

Solve the equation:

3x + 1 = 27 × 34

Solution 51

Question 52

Solve the equation:

Solution 52

Question 53

Solve the equation

3x-1 × 52y-3 = 225

Solution 53

Question 54

Solve the equation:

Solution 54

Question 55

Solve the equation:

Solution 55

Question 56

Solve the equation:

Solution 56

Question 57

Solution 57

Question 58

If a and b are different positive primes such that

Solution 58

Question 59

If a and b are different positive primes such that

(a + b)-1(a-1 + b-1) = axby, find x + y + 2.

Solution 59

Question 60

If 2x × 3y × 5z = 2160, find x, y and z. Hence, compute the value of 3x × 2-y × 5-z.

Solution 60

Question 61

If 1176 = 2a × 3b × 7c, find the values of a, b and c. hence, compute the value of 2a × 3b × 7-c as a fraction.

Solution 61

Question 62

Simplify :

Solution 62

Question 63

Simplify:

Solution 63

Question 64

Show that:

Solution 64

Question 65

If a = xm+nyl, b = xn+lym and c = xl+myn, prove that am-nbn-lcl-m = 1.

Solution 65

Question 66

If x = am+n, y = an+l and z = al+m, prove that xmynzl = znylzm.

Solution 66

Chapter 2 - Exponents of Real Numbers Excercise 2.29

Question 1

The value of {2 - 3(2 - 3)3}3 is

(a) 5

(b) 125

(c) 1/5

(d) -125

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).


Question 2

The value of x - yx-y when x = 2 and y = -2 is

(a) 18

(b) -18

(c) 14

(d) -14

Solution 2

x = 2, y = -2

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

Now x - yx-y = 2 - (-2)= 2 - 16 = -14

So, correct option is (d).

Question 3

The product of the square root of x with the cube root of x is

(a) cube root of the square root of x

(b) sixth root of the fifth power of x

(c) fifth root of the sixth power of x

(d) sixth root of x

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.

Question 4

The space seventh space root space of space straight x space divided space by space the space eighth space root space of space straight x space is
left parenthesis straight a right parenthesis space straight x
left parenthesis straight b right parenthesis space square root of straight x
left parenthesis straight c right parenthesis space root index 56 of straight x
left parenthesis straight d right parenthesis space fraction numerator 1 over denominator root index 56 of straight x end fraction

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.

Question 5

The space square space root space of space 64 space divided space by space the space cube space root space of space 64 space is
left parenthesis straight a right parenthesis space 64
left parenthesis straight b right parenthesis space 2
left parenthesis straight c right parenthesis space 1 half
left parenthesis straight d right parenthesis space 64 to the power of 2 divided by 3 end exponent

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 Excercise 2.30

Question 1

Which space of space the space following space is space left parenthesis are right parenthesis space not space equal space to 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 ?
left parenthesis straight a right parenthesis space open parentheses 5 over 6 close parentheses to the power of 1 fifth minus 1 over 6 end exponent
left parenthesis straight b right parenthesis space 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
left parenthesis straight c right parenthesis space open parentheses 6 over 5 close parentheses to the power of 1 divided by 30 end exponent
left parenthesis straight d right parenthesis space open parentheses 5 over 6 close parentheses to the power of negative 1 divided by 30 end exponent

Solution 1

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.

Question 2

When space simplified space 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 space is space equal space to
left parenthesis straight a right parenthesis space xy
left parenthesis straight b right parenthesis space straight x plus straight y
left parenthesis straight c right parenthesis space fraction numerator xy over denominator straight x plus straight y end fraction
left parenthesis straight d right parenthesis space fraction numerator straight x plus straight y over denominator xy end fraction

Solution 2

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.

Question 3

If 8x+1 = 64, what is the value of 32x+1?

(a) 1

(b) 3

(c) 9

(d) 27

Solution 3

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).

Question 4

If (23)2 = 4x, then 3x =

(a) 3

(b) 6

(c) 9

(d) 27

 

Solution 4

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.

Question 5

If x-2 = 64, then x1/3 + x0 =

(a) 2

(b) 3

(c) 3/2

(d) 2/3

Solution 5

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.

Question 6

When space simplified space open parentheses negative 1 over 27 close parentheses to the power of negative 2 divided by 3 end exponent space is
left parenthesis straight a right parenthesis space 9
left parenthesis straight b right parenthesis space minus 9
left parenthesis straight c right parenthesis space 1 over 9
left parenthesis straight d right parenthesis space minus 1 over 9

Solution 6

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.

Question 7

Which space one space of space the space following space is space not space equal space to space open parentheses cube root of 8 close parentheses to the power of negative 1 divided by 2 end exponent ?
left parenthesis straight a right parenthesis space open parentheses cube root of 2 close parentheses to the power of negative 1 divided by 2 end exponent
left parenthesis straight b right parenthesis space 8 to the power of negative 1 divided by 6 end exponent
left parenthesis straight c right parenthesis space 1 over open parentheses cube root of 8 close parentheses to the power of 1 divided by 2 end exponent
left parenthesis straight d right parenthesis space fraction numerator 1 over denominator square root of 2 end fraction

Solution 7

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.

Question 8

Which space one space of space the space following space is space not space equal space to space open parentheses 100 over 9 close parentheses to the power of negative 3 divided by 2 end exponent ?
left parenthesis straight a right parenthesis space open parentheses 9 over 100 close parentheses to the power of 3 divided by 2 end exponent
left parenthesis straight b right parenthesis space 1 over open parentheses begin display style 100 over 9 end style close parentheses to the power of 3 divided by 2 end exponent
left parenthesis straight c right parenthesis space 3 over 10 cross times 3 over 10 cross times 3 over 10
left parenthesis straight d right parenthesis space space square root of 100 over 9 cross times 100 over 9 cross times 100 over 9 end root

Solution 8

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.

Question 9

If space straight a comma space straight b comma space straight c space are space positive space real space numbers comma space then space square root of straight a to the power of negative 1 end exponent straight b end root cross times square root of straight b to the power of negative 1 end exponent straight c end root cross times square root of straight c to the power of negative 1 end exponent straight a end root space is space equal space to
left parenthesis straight a right parenthesis space 1
left parenthesis straight b right parenthesis space abc
left parenthesis straight c right parenthesis space square root of abc
left parenthesis straight d right parenthesis space 1 over abc

Solution 9

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.

Question 10

If space open parentheses 2 over 3 close parentheses to the power of straight x space open parentheses 3 over 2 close parentheses to the power of 2 straight x end exponent space equals 81 over 16 comma space then space straight x space equals
left parenthesis straight a right parenthesis space 2
left parenthesis straight b right parenthesis space 3
left parenthesis straight c right parenthesis space 4
left parenthesis straight d right parenthesis space 1

Solution 10

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 Excercise 2.31

Question 1

The space value space of space open curly brackets 8 to the power of negative 4 divided by 3 end exponent divided by 2 to the power of negative 2 end exponent close curly brackets to the power of 1 divided by 2 end exponent space is space
left parenthesis straight a right parenthesis space 1 half
left parenthesis straight b right parenthesis space 2
left parenthesis straight c right parenthesis space 1 fourth
left parenthesis straight d right parenthesis space 4

Solution 1

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.

Question 2

If space straight a comma space straight b comma space straight c space are space positive space real space numbers comma space then space fifth root of 3125 space straight a to the power of 10 straight b to the power of 5 straight c to the power of 10 end root space is space equal space to
left parenthesis straight a right parenthesis space 5 straight a squared bc squared
left parenthesis straight b right parenthesis space 25 ab squared straight c
left parenthesis straight c right parenthesis space 5 straight a cubed bc cubed
left parenthesis straight d right parenthesis space 125 straight a squared bc squared

Solution 2

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

Question 3

If space straight a comma space straight m comma space straight n space are space positive space integers comma space then space 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 space is space equal space to
left parenthesis straight a right parenthesis space straight a to the power of mn
left parenthesis straight b right parenthesis space straight a
left parenthesis straight c right parenthesis space straight a to the power of straight m divided by straight n end exponent
left parenthesis straight d right parenthesis space 1

Solution 3

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.

Question 4

If space straight x equals 2 space and space straight y equals 4 comma space then space 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 space equals space

Solution 4

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.

Question 5

The space value space of space straight m space for space which space open square brackets open curly brackets open parentheses 1 over 7 squared close parentheses to the power of negative 2 end exponent close curly brackets to the power of negative 1 divided by 3 end exponent close square brackets to the power of 1 divided by 4 end exponent space equals space 7 to the power of straight m space comma space is
left parenthesis straight a right parenthesis space fraction numerator negative 1 over denominator 3 end fraction
left parenthesis straight b right parenthesis space 1 fourth
left parenthesis straight c right parenthesis space minus 3
left parenthesis straight d right parenthesis space 2

Solution 5

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.

Question 6

The space value space of space open curly brackets open parentheses 23 plus 2 squared close parentheses to the power of 2 divided by 3 end exponent space plus space open parentheses 140 minus 19 close parentheses to the power of 1 divided by 2 end exponent close curly brackets squared comma space is
left parenthesis straight a right parenthesis space 196
left parenthesis straight b right parenthesis space 289
left parenthesis straight c right parenthesis space 324
left parenthesis straight d right parenthesis space 400

Solution 6

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.

Question 7

left parenthesis 256 right parenthesis to the power of 0.16 end exponent space cross times open parentheses 256 close parentheses to the power of 0.09 end exponent equals
left parenthesis straight a right parenthesis space 4
left parenthesis straight b right parenthesis space 16
left parenthesis straight c right parenthesis space 64
left parenthesis straight d right parenthesis space 256.25

Solution 7

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.

Question 8

If space 10 to the power of 2 straight y end exponent equals 25 comma space then space 10 to the power of negative straight y end exponent space equals
left parenthesis straight a right parenthesis space fraction numerator 1 over denominator negative 5 end fraction
left parenthesis straight b right parenthesis space 1 over 50
left parenthesis straight c right parenthesis space 1 over 625
left parenthesis straight d right parenthesis space 1 fifth

Solution 8

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.

Question 9

If 9x+2 = 240 + 9x, then x =

(a) 0.5

(b) 0.2

(c) 0.4

(d) 0.1

Solution 9

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.

Question 10

If x is a positive real number and x2 = 2, then x3 =

left parenthesis straight a right parenthesis space square root of 2
left parenthesis straight b right parenthesis space 2 square root of 2
left parenthesis straight c right parenthesis space 3 square root of 2
left parenthesis straight d right parenthesis space 4

Solution 10

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

Question 11

If space 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 space and space straight x space greater than space 0 comma space then space straight x space equals
left parenthesis straight a right parenthesis space fraction numerator square root of 2 over denominator 4 end fraction
left parenthesis straight b right parenthesis space 2 square root of 2
left parenthesis straight c right parenthesis space 4
left parenthesis straight d right parenthesis space 64

Solution 11

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 Excercise 2.32

Question 1

If space 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 comma space what space is space the space value space of space straight g space when space straight t space equals space 64 ?
left parenthesis straight a right parenthesis space 31 over 2
left parenthesis straight b right parenthesis space 33 over 2
left parenthesis straight c right parenthesis space 16
left parenthesis straight d right parenthesis space 257 over 16

Solution 1

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.

Question 2

If space 4 to the power of straight x minus space 4 to the power of straight x minus 1 end exponent space equals space 24 comma space then space left parenthesis 2 straight x right parenthesis to the power of straight x space equal
left parenthesis straight a right parenthesis space 5 square root of 5
left parenthesis straight b right parenthesis space square root of 5
left parenthesis straight c right parenthesis space 25 square root of 5
left parenthesis straight d right parenthesis space 125

 

Solution 2

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.

Question 3

When space simplified space 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 space is
left parenthesis straight a right parenthesis space 8
left parenthesis straight b right parenthesis space 1 over 8
left parenthesis straight c right parenthesis space 2
left parenthesis straight d right parenthesis space 1 half

Solution 3

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.

Question 4

If space 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 comma space then space straight x space equals
left parenthesis straight a right parenthesis space 2
left parenthesis straight b right parenthesis space 3
left parenthesis straight c right parenthesis space 5
left parenthesis straight d right parenthesis space 4

Solution 4

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.

Question 5

The space value space of space 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 comma space is
left parenthesis straight a right parenthesis space 1
left parenthesis straight b right parenthesis space 1 third
left parenthesis straight c right parenthesis space minus 3
left parenthesis straight d right parenthesis space minus 2

Solution 5

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

Question 6

If space square root of 5 to the power of straight n end root equals 125 comma space then space 5 to the power of straight n-th root of 64 end exponent equals
left parenthesis straight a right parenthesis space 25
left parenthesis straight b right parenthesis space 1 over 125
left parenthesis straight c right parenthesis space 625
left parenthesis straight d right parenthesis space 1 fifth

Solution 6

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.

Question 7

If (16)2x+3 = (64)x+3, then 42x-2 =

(a) 64

(b) 256

(c) 32

(d) 512

Solution 7

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.

Question 8

If space 2 to the power of negative straight m end exponent cross times 1 over 2 to the power of straight m equals 1 fourth comma space then space 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 plus open parentheses 1 over 5 to the power of straight m close parentheses to the power of negative 1 end exponent close curly brackets space is space equal space to
left parenthesis straight a right parenthesis space 1 half
left parenthesis straight b right parenthesis space 2
left parenthesis straight c right parenthesis space 4
left parenthesis straight d right parenthesis space minus 1 fourth

Solution 8

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.

Question 9

If space 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 space comma space 3 to the power of straight p over 3 to the power of straight n space equals space 81 space and space straight a space equals space 2 to the power of 1 divided by 10 end exponent space comma space then 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 space equals
left parenthesis straight a right parenthesis space 2
left parenthesis straight b right parenthesis space 1 fourth
left parenthesis straight c right parenthesis space 9
left parenthesis straight d right parenthesis space 1 over 8

Solution 9

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.

Question 10

If space 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 comma space then space straight x space equals
left parenthesis straight a right parenthesis space 3
left parenthesis straight b right parenthesis space minus 3
left parenthesis straight c right parenthesis space 1 third
left parenthesis straight d right parenthesis space minus 1 third

Solution 10

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 Excercise 2.33

Question 1

If space 0 space less than space straight y space less than space straight x comma space which space statement space must space be space true ?
left parenthesis straight a right parenthesis space square root of straight x minus square root of straight y equals square root of straight x minus straight y end root
left parenthesis straight b right parenthesis space square root of straight x plus square root of straight x equals square root of 2 straight x end root
left parenthesis straight c right parenthesis space space straight x square root of straight y equals straight y square root of straight x
left parenthesis straight d right parenthesis space square root of xy equals square root of straight x square root of straight y

Solution 1

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.

Question 2

If space 10 to the power of straight x space equals space 64 comma space w hat space is space the space value space of space 10 to the power of straight x over 2 plus 1 end exponent ?
left parenthesis straight a right parenthesis space 18
left parenthesis straight b right parenthesis space 42
left parenthesis straight c right parenthesis space 80
left parenthesis straight d right parenthesis space 81

Solution 2

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.

Question 3

fraction numerator 5 to the power of straight n plus 2 end exponent space minus space 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 is space equal space to
left parenthesis straight a right parenthesis space 5 over 3
left parenthesis straight b right parenthesis space minus 5 over 3
left parenthesis straight c right parenthesis space 3 over 5
left parenthesis straight d right parenthesis space minus 3 over 5

Solution 3

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.

Question 4

If space square root of 2 to the power of straight n end root space equals space 1024 comma space then space 3 to the power of 2 open parentheses straight n over 4 minus 4 close parentheses end exponent space equals
left parenthesis straight a right parenthesis space 3
left parenthesis straight b right parenthesis space 9
left parenthesis straight c right parenthesis space 27
left parenthesis straight d right parenthesis space 81

Solution 4

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.

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