Class 8 RD Sharma Solutions – Chapter 2 Powers – Exercise 2.2 | Set 1

Question 1. Write each of the following in exponential form:

(i) (3/2)^-1 × (3/2)^-1 × (3/2)^-1 × (3/2)^-1

Solution:

= (3/2)^-1 × (3/2)^-1 × (3/2)^-1 × (3/2)^-1

= (3/2)^-4 (aⁿ × a^m = a^{n + m})

(ii) (2/5)^-2 × (2/5)^-2 × (2/5)^-2

Solution:

= (2/5)^-2 × (2/5)^-2 × (2/5)^-2

= (2/5)^-6 (aⁿ × a^m = a^{n + m})

Question 2. Evaluate:

(i) 5^-2

Solution:

= 5^-2

= 1/5² (a^-n = 1/aⁿ)

= 1/25 

(ii) (-3)^-2

Solution:

= (-3)^-2

= (1/-3)² (a^-n = 1/aⁿ)

= 1/9 

(iii) (1/3)^-4

Solution:

= (1/3)^-4

= 3⁴ (a^-n = 1/aⁿ)

= 81

(iv) (-1/2)^-1

Solution:

= (-1/2)^-1

= -2¹ (a^-n = 1/aⁿ)

= -2

Question 3. Express each of the following as a rational number in the form p/q:

(i) 6^-1

Solution:

= 6^-1

= 1/6¹ = 1/6 (a^-n = 1/aⁿ)

(ii) (-7)^-1

Solution:

= (-7)^-1

= 1/-7¹ (a^-n = 1/aⁿ)

= -1/7 

(iii) (1/4)^-1

Solution:

= (1/4)^-1

= 4¹ (1/a^-n = aⁿ)

= 4 

(iv) (-4)^-1 × (-3/2)^-1

Solution:

= (-4)^-1 × (-3/2)^-1

= 1/-4¹ × (2/-3)¹ (a^-n = 1/aⁿ, 1/a^-n = aⁿ)

2 is the common factor

= 1/-2 × -1/3

= 1/6

(v) (3/5)^-1 × (5/2)^-1

Solution:

= (3/5)^-1 × (5/2)^-1(a^-n = 1/aⁿ⁾

= (5/3)¹ × (2/5)¹ 

= 5/3 × 2/5

= 2/3

Question 4. Simplify:

(i) (4^-1 × 3^-1)²

Solution:

= (4^-1 × 3^-1)²

= (1/4 × 1/3)² (a^-n = 1/aⁿ)

= (1/12)²

= 1/144

(ii) (5^-1 ÷ 6^-1)³

Solution:

= (5^-1 ÷ 6^-1)³

= (1/5 ÷ 1/6)³ (a^-n = 1/aⁿ)

= (1/5 × 6)³

= (6/5)³

= 216/125

(iii) (2^-1 + 3^-1)^-1

Solution:

= (2^-1 + 3^-1)^-1

= (1/2 + 1/3)^-1 (a^-n = 1/aⁿ)

LCM of 2 and 3 is 6

= ((3+2)/6)^-1

= (5/6)^-1 (1/a^-n = aⁿ)

= 6/5

(iv) (3^-1 × 4^-1)^-1 × 5^-1

Solution:

= (3^-1 × 4^-1)^-1 × 5^-1

= (1/3 × 1/4)^-1 × 1/5 (a^-n = 1/aⁿ)

= (1/12)^-1 × 1/5 (1/a^-n = aⁿ)

= 12 × 1/5

= 12/5

(v) (4^-1 – 5^-1) ÷ 3^-1

Solution:

= (4^-1 – 5^-1) ÷ 3^-1

= (1/4 – 1/5) ÷ 1/3 (a^-n = 1/aⁿ)

LCM of 4 and 5 is 20

= (5 – 4)/20 × 3/1 

= 1/20 × 3

= 3/20

Question 5. Express each of the following rational numbers with a negative exponent:

(i) (1/4)³

Solution:

= (1/4)³

= (4)^-3 (1/aⁿ = a^-n)

(ii)3⁵

Solution:

= 3⁵

= (1/3)^-5 (1/aⁿ = a^-n)

(iii) (3/5)⁴

Solution:

= (3/5)⁴

= (5/3)^-4 (a/b)^-n = (b/a)ⁿ

(iv) ((3/2)⁴)^-3

Solution:

= ((3/2)⁴)^-3

= (3/2)^-12 ((aⁿ)^m = a^nm)

(v) ((7/3)⁴)^-3

Solution:

= ((7/3)⁴)^-3

= (7/3)^-12 ((aⁿ)^m = a^nm)

Question 6. Express each of the following rational numbers with a positive exponent:

(i) (3/4)^-2

Solution:

= (3/4)^-2

= (4/3)² ((a/b)^-n = (b/a)ⁿ)

(ii) (5/4)^-3

Solution:

= (5/4)^-3

= (4/5)³ ((a/b)^-n = (b/a)ⁿ)

(iii) 4³ × 4^-9

Solution:

= 4³ × 4^-9

= (4)^{3 – 9} (aⁿ × a^m = a^{n + m})

= 4^-6

= (1/4)⁶ (1/aⁿ = a^-n)

(iv) ((4/3)^-3)^-4

Solution:

= ((4/3)^-3)^-4

= (4/3)¹² ((aⁿ)^m = a^nm)

(v) ((3/2)⁴)^-2

Solution:

= ((3/2)⁴)^-2

= (3/2)^-8 ((aⁿ)^m = a^nm)

= (2/3)⁸ (1/aⁿ = a^-n)

Question 7. Simplify:

(i) ((1/3)^-3 – (1/2)^-3) ÷ (1/4)^-3

Solution:

= ((1/3)^-3 – (1/2)^-3) ÷ (1/4)^-3

= (3³ – 2³) ÷ 4³ (1/aⁿ = a^-n)

= (27-8) ÷ 64

= 19 ÷ 64

= 19/64

(ii) (3² – 2²) × (2/3)^-3

Solution:

= (3² – 2²) × (2/3)^-3

= (9 – 4) × (3/2)³ (1/aⁿ = a^-n)

= 5 × (27/8)

= 135/8

(iii) ((1/2)^-1 × (-4)^-1)^-1

Solution:

= ((1/2)^-1 × (-4)^-1)^-1

= (2¹ × (1/-4))^-1 (1/aⁿ = a^-n)

2 is the common factor

= (1/-2)^-1 (1/aⁿ = a^-n)

= -2¹

= -2

(iv) (((-1/4)²)^-2)^-1

Solution:

= (((-1/4)²)^-2)^-1

= ((1/16)^-2)^-1 (1/aⁿ = a^-n)

= ((16)²)^-1 (1/aⁿ = a^-n)

= (256)^-1 (1/aⁿ = a^-n)

= 1/256

(v) ((2/3)²)³ × (1/3)^-4 × 3^-1 × 6^-1

Solution:

= ((2/3)²)³ × (1/3)^-4 × 3^-1 × 6^-1

= (4/9)³ × 3⁴ × 1/3 × 1/6 (1/aⁿ = a^-n)

= (64/729) × 81 × 1/3 × 1/6

3 is the common factor

= (64/729) × 27 × 1/6

= 32/729 × 27 × 1/3

3 is the common factor

= 32/729 × 9

9 is the common factor

= 32/81

Question 8. By what number should 5^-1 be multiplied so that the product may be equal to (-7)^-1?

Solution:

Let the number be x

 5^-1 × x = (-7)^-1

1/5 × x = 1/-7 (1/aⁿ = a^-n)

x = (-1/7) / (1/5)

  = (-1/7) × (5/1) 

  = -5/7

It should be multiplied with -5/7

Question 9. By what number should (1/2)^-1 be multiplied so that the product may be equal to (-4/7)^-1?

Solution:

Let the number be x

 (1/2)^-1 × x = (-4/7)^-1

1/(1/2) × x = 1/(-4/7) (we know that 1/aⁿ = a^-n)

x = (-7/4) / (2/1)

  = (-7/4) × (1/2)

  = -7/8

It should be multiplied with -7/8

Question 10. By what number should (-15)^-1 be divided so that the quotient may be equal to (-5)^-1?

Solution:

Let the number be x

So, (-15)^-1 ÷ x = (-5)^-1 (we know that 1/a ÷ 1/b = 1/a × b/1)

1/-15 × 1/x = 1/-5 (we know that 1/aⁿ = a^-n)

1/x = (1× – 15)/-5

1/x = 3

x = 1/3

It should be divided by 1/3

Chapter 2 Powers – Exercise 2.2 | Set 2