Class 8 RD Sharma Solutions Chapter 2 Powers Exercise 2.2 | Set 1 (original) (raw)
Last Updated : 23 Jul, 2025
In mathematics, powers are a fundamental concept used to the express repeated multiplication of a number by itself. In Chapter 2 of RD Sharma's Class 8 textbook, we explore how to work with the powers including how to simplify expressions involving the exponents and solve problems that utilize these principles. This chapter lays the foundation for understanding more advanced topics in algebra and helps students develop essential problem-solving skills.
Powers
A power is represented as an where a is the base and n is the exponent. The exponent n indicates how many times the base a is multiplied by itself. For example, 34 equals 3 × 3 × 3 × 3 = 81. Understanding powers involves learning the basic laws of the exponents such as:
- Multiplying Powers with the Same Base: a^m \times a^n = a^{m+n}
- Dividing Powers with the Same Base:\frac{a^m}{a^n} = a^{m-n}
- Power of a Power:(a^m)^n = a^{m \times n}
Exercise 2.2 | Set 1 of Chapter 2 (Powers) in RD Sharma's Class 8 mathematics textbook builds upon the foundational concepts introduced in Exercise 2.1. This section delves deeper into the application of exponent laws, focusing on more complex problems that require a combination of multiple rules. Students are challenged to simplify expressions involving fractional exponents, negative bases, and mixed operations. The problems in this set are designed to enhance critical thinking and problem-solving skills, encouraging students to apply their knowledge of exponent laws in varied contexts. This exercise plays a crucial role in preparing students for advanced algebraic concepts and lays the groundwork for understanding higher-level mathematical operations involving powers and radicals.
**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 × am = an + 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 × am = an + 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)3** 5
**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 = anm)
****(v) ((7/3)** 4 ) -3
**Solution:
= ((7/3)4)-3
= (7/3)-12 ((an)m = anm)
**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) 4** 3 × 4 -9
**Solution:
= 43 × 4-9
= (4)3 - 9 (an × am = an + 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 = anm)
****(v)** ****((3/2)** 4 ) -2
**Solution:
= ((3/2)4)-2
= (3/2)-8 ((an)m = anm)
= (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) (3** 2 - 2 2 ) × (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
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Summary
Exercise 2.2 | Set 1 of Chapter 2 in RD Sharma's Class 8 mathematics textbook expands on the fundamental exponent laws, presenting students with more challenging and diverse problems. This set focuses on applying multiple exponent rules simultaneously, working with fractional and negative exponents, and manipulating expressions with negative bases. The problems are carefully structured to enhance students' ability to break down complex expressions, apply appropriate exponent laws, and arrive at simplified solutions. Through this exercise, students develop a deeper understanding of the relationships between different types of exponents and their operations. The skills honed in this set are essential for tackling more advanced mathematical concepts in algebra, calculus, and other areas of mathematics. By mastering these problems, students build confidence in handling intricate mathematical expressions and develop problem-solving strategies that will serve them well in future mathematical endeavors.