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# 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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