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Class 8 RD Sharma Solutions – Chapter 2 Powers – Exercise 2.2 | Set 1
  • Last Updated : 10 Nov, 2020

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)

= 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 = 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) 43 × 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) (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

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