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

Chapter 2 of RD Sharma's Class 8 mathematics textbook focuses on Powers. Exercise 2.1 specifically deals with the laws of exponents and their applications. This section helps students understand how to manipulate expressions involving powers and use exponent rules to simplify mathematical expressions. Mastering these co

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

Summary

Exercise 2.1 of Chapter 2 (Powers) in RD Sharma's Class 8 mathematics textbook is a comprehensive exploration of the fundamental laws of exponents. This exercise introduces students to key concepts such as the product rule, quotient rule, power of a power rule, and the handling of zero and negative exponents. Through a series of carefully crafted problems, students learn to apply these rules to simplify complex expressions, solve equations involving exponents, and evaluate powers with various bases and exponents. The exercise builds a strong foundation in working with powers, which is crucial for more advanced algebraic concepts. By mastering the content of this exercise, students develop the skills needed to manipulate expressions involving powers efficiently, a capability that proves invaluable in higher-level mathematics and real-world applications. The problems in this exercise range from basic computations to more challenging scenarios, providing a well-rounded understanding of exponent manipulation and its practical uses.