**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}= 1/2^{3}= 1/2×2×2 = 1/8 (we know that a^{-n}= 1/a^{n})

**(ii) (-4) ^{-2}**

**Solution:**

(-4)

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

**(iii) 1/(3) ^{-2}**

**Solution:**

1/(3)

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

**(iv) (1/2) ^{-5}**

**Solution:**

(1/2)

^{-5}= 2^{5 }/ 1^{5}= 2 × 2 × 2 × 2 × 2 = 32 (we know that a^{-n}= 1/a^{n})

**(v) (2/3) ^{-2}**

**Solution:**

(2/3)

^{-2}= 3^{2}/ 2^{2}= (3 × 3) / (2 × 2) = 9/4 (we know that a^{-n}= 1/a^{n})

**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/a^{n})LCM of 3 and 4 is 12

= (1 × 4 + 1 × 3) / 12

= (4 + 3) / 12

= 7/12

**(ii) (3 ^{0} + 4^{-1}) × 2^{2}**

**Solution:**

= (3

^{0}+ 4^{-1}) × 2^{2}= (1 + 1/4) × 4 (we know that a

^{-n}= 1/a^{n}, a^{0}= 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 a^{0}= 1)

**(iv) ((1/3) ^{-1} – (1/4)^{-1})^{-1}**

**Solution:**

= ((1/3)

^{-1}– (1/4)^{-1})^{-1}= (3

^{1}– 4^{1})^{-1}(1/a^{-n}= a^{n}, a^{-n}= 1/a^{n})= (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}= 2

^{1}+ 3^{1}+ 4^{1}(1/a^{-n}= a^{n})= 2 + 3 + 4

= 9

**(ii) (1/2) ^{-2} + (1/3)^{-2} + (1/4)^{-2}**

**Solution:**

= (1/2)

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

^{2}+ 3^{2}+ 4^{2}(1/a^{-n}= a^{n})= 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) / (1/2

^{2}) (a^{-n}= 1/a^{n})= (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}= (1/5

^{1}× 1/2^{1}) / (1/6^{1}) (a^{-n}= 1/a^{n})= (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/a^{n})= (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/a^{n})= ((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/a^{n})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/a^{n})= (1/12)

^{-1}× 1/5=12 × 1/5

= 12/5

**Question 5. Simplify:**

**(**i**) (3 ^{2} + 2^{2}) × (1/2)^{3}**

**Solution:**

= (3

^{2}+ 2^{2}) × (1/2)^{3}= (9 + 4) × 1/8

= 13/8

**(ii) (3 ^{2} – 2^{2}) × (2/3)^{-3}**

**Solution:**

= (3

^{2}– 2^{2}) × (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}= (3

^{3 }– 2^{3}) ÷ 4^{3}(1/a^{-n}= a^{n})= (27 – 8) ÷ 64

= 19/64

**(iv) (2 ^{2} + 3^{2} – 4^{2}) ÷ (3/2)^{2}**

**Solution:**

= (2

^{2}+ 3^{2}– 4^{2}) ÷ (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}?

^{-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}?

^{-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}?

^{-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