**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}(a^{n}× a^{m}= a^{n + m})

**(ii) (2/5) ^{-2} × (2/5)^{-2} × (2/5)^{-2}**

**Solution:**

= (2/5)

^{-2}× (2/5)^{-2}× (2/5)^{-2}= (2/5)

^{-6}(a^{n}× a^{m}= a^{n + m})

**Question 2. Evaluate:**

**(i) 5 ^{-2}**

**Solution:**

= 5

^{-2}= 1/5

^{2}(a^{-n}= 1/a^{n})= 1/25

**(ii) (-3) ^{-2}**

**Solution:**

= (-3)

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

^{2}(a^{-n}= 1/a^{n})= 1/9

**(iii) (1/3) ^{-4}**

**Solution:**

= (1/3)

^{-4}= 3

^{4}(a^{-n}= 1/a^{n})= 81

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

**Solution:**

= (-1/2)

^{-1}= -2

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

**Question 3. Express each of the following as a rational number in the form p/q:**

**(i) 6 ^{-1}**

**Solution:**

= 6

^{-1}= 1/6

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

**(ii) (-7) ^{-1}**

**Solution:**

= (-7)

^{-1}= 1/-7

^{1}(a^{-n}= 1/a^{n})= -1/7

**(iii) (1/4) ^{-1}**

**Solution:**

= (1/4)

^{-1}= 4

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

**(iv) (-4) ^{-1} × (-3/2)^{-1}**

**Solution:**

= (-4)

^{-1}× (-3/2)^{-1}= 1/-4

^{1}× (2/-3)^{1}(a^{-n}= 1/a^{n}, 1/a^{-n}= a^{n})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/a^{n)}= (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/a^{n})= (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/a^{n})= (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/a^{n})LCM of 2 and 3 is 6

= ((3+2)/6)

^{-1}= (5/6)

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

**(ii)3 ^{5}**

**Solution:**

= 3

^{5}=

^{ }(1/3)^{-5}(1/a^{n}= 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 }((a^{n})^{m}= a^{nm})

**(v) ((7/3) ^{4})^{-3}**

**Solution:**

= ((7/3)

^{4})^{-3}= (7/3)

^{-12}((a^{n})^{m}= a^{nm})

**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:**

= 4

^{3}× 4^{-9}= (4)

^{3 – 9}(a^{n}× a^{m}= a^{n + m})= 4

^{-6}= (1/4)

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

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

**Solution:**

= ((4/3)

^{-3})^{-4}= (4/3)

^{12 }((a^{n})^{m}= a^{nm})

**(v)** **((3/2) ^{4})^{-2}**

**Solution:**

= ((3/2)

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

^{-8}((a^{n})^{m}= a^{nm})= (2/3)

^{8}(1/a^{n}= 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}= (3

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

= 19 ÷ 64

= 19/64

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

**Solution:**

= (3

^{2}– 2^{2}) × (2/3)^{-3}= (9 – 4) × (3/2)

^{3}(1/a^{n}= a^{-n})= 5 × (27/8)

= 135/8

**(iii)** **((1/2) ^{-1} × (-4)^{-1})^{-1}**

**Solution:**

= ((1/2)

^{-1}× (-4)^{-1})^{-1}= (2

^{1}× (1/-4))^{-1}(1/a^{n}= a^{-n})2 is the common factor

= (1/-2)

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

^{1}= -2

**(iv) (((-1/4) ^{2})^{-2})^{-1}**

**Solution:**

= (((-1/4)

^{2})^{-2})^{-1}= ((1/16)

^{-2})^{-1}(1/a^{n}= a^{-n})= ((16)

^{2})^{-1}(1/a^{n}= a^{-n})= (256)

^{-1}(1/a^{n}= 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}× 3^{4}× 1/3 × 1/6 (1/a^{n}= 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}?

^{-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/a

^{n}= 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}?

^{-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/a

^{n}= 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}?

^{-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/a

^{n}= a^{-n})1/x = (1× – 15)/-5

1/x = 3

x = 1/3

It should be divided by 1/3