Class 8 RD Sharma Solutions – Chapter 2 Powers – Exercise 2.1

Question 1. Express each of the following as a rational number of the form p/q, where p and q are integers and q ≠ 0

(i) 2^-3

Solution:

2^-3 = 1/2³ = 1/2×2×2 = 1/8 (we know that a^-n = 1/aⁿ)

(ii) (-4)^-2

Solution:

(-4)^-2 = 1/-4² = 1/-4×-4 = 1/16 (we know that a^-n = 1/aⁿ)

(iii) 1/(3)^-2

Solution:

1/(3)^-2 = 3² = 3 × 3 = 9 (we know that 1/a^-n = aⁿ)

(iv) (1/2)^-5

Solution:

(1/2)^-5 = 2⁵ / 1⁵ = 2 × 2 × 2 × 2 × 2 = 32 (we know that a^-n = 1/aⁿ)

(v) (2/3)^-2

Solution:

(2/3)^-2 = 3² / 2² = (3 × 3) / (2 × 2) = 9/4 (we know that a^-n = 1/aⁿ)

Question 2. Find the values of each of the following:

(i) 3^-1 + 4^-1

Solution:

= 3^-1 + 4^-1

= 1/3 + 1/4 (we know that a^-n = 1/aⁿ)

LCM of 3 and 4 is 12

= (1 × 4 + 1 × 3) / 12

= (4 + 3) / 12

= 7/12

(ii) (3⁰ + 4^-1) × 2²

Solution:

= (3⁰ + 4^-1) × 2²

= (1 + 1/4) × 4 (we know that a^-n = 1/aⁿ, a⁰ = 1)

LCM of 1 and 4 is 4

= (1 × 4 + 1 × 1) / 4 × 4

= (4 + 1) / 4 × 4

= 5/4 × 4

= 5

(iii) (3^-1 + 4^-1 + 5^-1)⁰

Solution:

(3^-1 + 4^-1 + 5^-1)⁰ = 1 (We know that a⁰ = 1)

(iv) ((1/3)^-1 – (1/4)^-1)^-1

Solution:

= ((1/3)^-1 – (1/4)^-1)^-1

= (3¹ – 4¹)^-1 (1/a^-n = aⁿ, a^-n = 1/aⁿ)

= (3 – 4)^-1

= (-1)^-1

= 1/-1 = -1

Question 3. Find the values of each of the following:

(i) (1/2)^-1 + (1/3)^-1 + (1/4)^-1

Solution:

= (1/2)^-1 + (1/3)^-1 + (1/4)^-1

= 2¹ + 3¹ + 4¹ (1/a^-n = aⁿ)

= 2 + 3 + 4

= 9

(ii) (1/2)^-2 + (1/3)^-2 + (1/4)^-2

Solution:

= (1/2)^-2 + (1/3)^-2 + (1/4)^-2

= 2² + 3² + 4² (1/a^-n = aⁿ)

= 2 × 2 + 3 × 3 + 4 × 4

= 4 + 9 + 16 

= 29

(iii) (2^-1 × 4^-1) ÷ 2^-2

Solution:

= (2^-1 × 4^-1) ÷ 2^-2

= (1/2 × 1/4) / (1/2²) (a^-n = 1/aⁿ)

= (1/2 × 1/4) × 4/1

= 1/8 × 4/1

4 is the common factor

= 1/2

(iv) (5^-1 × 2^-1) ÷ 6^-1

Solution:

= (5^-1 × 2^-1) ÷ 6^-1

= (1/5¹ × 1/2¹) / (1/6¹) (a^-n = 1/aⁿ)

= (1/5 × 1/2) × 6/1

= 1/10 × 6/1

2 is the common factor

= 3/5

Question 4. Simplify:

(i) (4^-1 × 3^-1)²

Solution:

= (4^-1 × 3^-1)² (a^-n = 1/aⁿ)

= (1/4 × 1/3)²

= (1/12)²

= (1 × 1 / 12 × 12)

= 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

= 6 × 6 × 6 / 5 × 5 × 5

= 216/125

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

Solution:

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

= (1/2 + 1/3)^-1 (we know that a^-n = 1/aⁿ)

LCM of 2 and 3 is 6

= ((1 × 3 + 1 × 2)/6)^-1

= (5/6)^-1

= 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

=12 × 1/5

= 12/5

Question 5. Simplify:

(i) (3² + 2²) × (1/2)³

Solution:

= (3² + 2²) × (1/2)³

= (9 + 4) × 1/8 

= 13/8

(ii) (3² – 2²) × (2/3)^-3

Solution:

= (3² – 2²) × (2/3)^-3

= (9 – 4) × (3/2)³

= 5 × (27/8)

= 135/8

(iii) ((1/3)^-3 – (1/2)^-3) ÷ (1/4)^-3

Solution:

= ((1/3)^-3 – (1/2)^-3) ÷ (1/4)^-3

= (3³ – 2³) ÷ 4³ (1/a^-n = aⁿ)

= (27 – 8) ÷ 64

= 19/64

(iv) (2² + 3² – 4²) ÷ (3/2)²

Solution:

= (2² + 3² – 4²) ÷ (3/2)²

= (4 + 9 – 16) ÷ (9/4)

= (13 – 16) / 9/4

= (-3) × 4/9 

3 is the common factor

= -4/3

Question 6. 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

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

= (-1/7) × (5/1)

= -5/7

It should be multiplied with -5/7

Question 7. 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)

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

= (-7/4) × (1/2)

= -7/8

It should be multiplied with -7/8

Question 8. By what number should (-15)^-1 be divided so that the quotient may be equal to (-5)^-1?

Solution:

Let the number be x

(-15)^-1 ÷ x = (-5)^-1

1/-15 × 1/x = 1/-5

1/x = (1× – 15) / -5

1/x = 3

x = 1/3

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