# Solved Examples on Exponents

**Here are some solved examples on exponents using the laws of exponents.**

1. Evaluate the exponent:

(i) 5

^{-3}
(ii) (

^{1}/

_{3})

^{-4}
(iii) (

^{5}/

_{2})

^{-3}
(iv) (-2)

^{-5}
(v) (

^{-3}/

_{4})

^{-4}
**We have: **
** (i) **5

^{-3} = 1/5

^{3} = 1/125

** (ii) ** (1/3)

^{-4} = (3/1)

^{4} = 3

^{4} = 81

** (iii) ** (5/2)

^{-3} = (2/5)

^{3} = 2

^{3}/5

^{3} = 8/125

** (iv) ** (-2)

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

^{-5} = 1/-2

^{5} = 1/-32 = -1/32

**(v)** (-3/4)

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

^{4} = (-4/3)

^{4} = (-4)

^{4}/3

^{4} = 4

^{4}/3

^{4} = 256/81

**2. Evaluate: (**^{-2}/_{7})^{-4} × (^{-5}/_{7})^{2}
**Solution:**
(

^{-2}/

_{7})

^{-4} × (

^{-5}/

_{7})

^{2}
= (7/-2)

^{4} × (-5/7)

^{2}
= (-7/2)

^{4} × (-5/7)

^{2} **[Since, (7/-2) = (-7/2)]**
= (-7)

^{4}/2

^{4} × (-5)

^{2}/7

^{2}
= {7

^{4} × (-5)

^{2}}/{2

^{4} × 7

^{2} }

**[Since, (-7)**^{4} = 7^{4}]
= {7

^{2} × (-5)

^{2} }/2

^{4}
= [49 × (-5) × (-5)]/16

= 1225/16

**3. Evaluate: (-1/4)**^{-3} × (-1/4)^{-2}

Solution:
(-1/4)

^{-3} × (-1/4)

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

^{3} × (4/-1)

^{2}
= (-4)

^{3} × (-4)

^{2}
= (-4)

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

^{5}
= -4

^{5}
= -1024.

**4. Evaluate: {[(-3)/2]**^{2}}^{-3}

Solution:
{[(-3)/2]

^{2}}

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

^{2 × (-3)}
= (-3/2)

^{-6}
= (2/-3)

^{6}
= (-2/3)

^{6}
= (-2)

^{6}/3

^{6}
= 2

^{6}/3

^{6}
= 64/729

**5. Simplify: **
(i) (2

^{-1} × 5

^{-1})

^{-1} ÷ 4

^{-1}
(ii) (4

^{-1} + 8

^{-1}) ÷ (2/3)

^{-1}
**Solution:**
(i) (2

^{-1} × 5

^{-1})

^{-1} ÷ 4

^{-1}
= (1/2 × 1/5)

^{-1} ÷ (4/1)

^{-1}
= (1/10)

^{-1} ÷ (1/4)

=

^{10}/

_{1} ÷

^{1}/

_{4}
= (10 ÷

^{1}/

_{4})

= (10 × 4)

= 40.

** (ii) (4**^{-1} + 8^{-1}) ÷ (2/3)^{-1}
= (1/4 + 1/8) ÷ (3/2)

= (2 + 1)/8 ÷ 3/2

= (3/8 ÷ 3/2)

= (3/8 ÷ 2/3)

= 1/4

**6. Simplify: (1/2)**^{-2} + (1/3)^{-2} + (1/4)^{-2}

Solution:
(1/2)

^{-2} + (1/3)

^{-2} + (1/4)

^{-2}
= (2/1)

^{2} + (3/1)

^{2} + (4/1)

^{2}
= (2

^{2} + 3

^{2} + 4

^{2})

= (4 + 9 + 16)

= 29.

**7. By what number should (1/2)**^{-1} be multiplied so that the product is (-5/4)^{-1}?

Solution:
Let the required number be x. Then,

x × (1/2)

^{-1} = (-5/4)

^{-1}
⇒ x × (2/1) = (4/-5)

⇒ 2x = -4/5

⇒ x = (1/2 × -4/5) = -2/5

**Hence, the required number is -2/5. **

8. By what number should (-3/2)^{-3} be divided so that the quotient is (9/4)^{-2}?

Solution:
Let the required number be x. Then,

(-3/2)

^{-3}/x = (9/4)

^{-2}
⇒ (-2/3)

^{3} = (4/9)

^{2} × x

⇒ (-2)

^{3}/3

^{3} = 4

^{2}/9

^{2} × x

⇒ -8/27 = 16/81 × x

⇒ x = {-8/27 × 81/16}

⇒ x = -3/2

**Hence, the required number is -3/2 **
**9. If a = (2/5)**^{2} ÷ (9/5)^{0} find the value of a^{-3}.

Solution:
a

^{-3} = [(2/5)

^{2} ÷ (9/5)

^{0}]

^{-3}
= [(2/5)

^{2} ÷ 1]

^{-3}
= [(2/5)

^{2}]

^{-3}
= (2/5)

^{-6}
= (5/2)

^{6}
**10. Find the value of n, when 3**^{-7} ×3^{2n + 3} = 3^{11} ÷ 3^{5}

Solution:
3

^{2n + 3} = 3

^{11} ÷ 3

^{5}/3

^{-7}
⇒ 3

^{2n + 3} = 3

^{11 - 5}/3

^{-7}
⇒ 3

^{2n + 3} = 3

^{6}/3

^{-7}
⇒ 3

^{2n + 3} = 3

^{6 - (-7)}
⇒ 3

^{2n + 3} = 3

^{6 + 7}
⇒ 3

^{2n + 3} = 3

^{13}
Since the bases are same and equating the powers, we get 2n + 3 = 13

2n = 13 – 3

2n = 10

n = 10/2

**Therefore, n = 5**

11. Find the value of n, when (5/3)^{2n + 1} (5/3)^{5} = (5/3)^{n + 2}

Solution:
(5/3)

^{2n + 1 + 5} = (5/3)

^{n + 2}
= (5/3)

^{2n + 6} = (5/3)

^{n + 2}
Since the bases are same and equating the powers, we get 2n + 6 = n + 2

2n – n = 2 – 6

=> n = -4

**12. Find the value of n, when 3**^{n} = 243

Solution:
3

^{n} = 3

^{5}
Since, the bases are same, so omitting the bases, and equating the powers we get, n = 5.

**13. Find the value of n, when 27**^{1/n} = 3

Solution:
(27) = 3

^{n}
⇒ (3)

^{3} = 3

^{n}
Since, the bases are same and equating the powers, we get

⇒ n = 3

**14. Find the value of n, when 343**^{2/n} = 49

Solution:
[(7)

^{3}]

^{2/n} = (7)

^{2}
⇒ (7)

^{6/n} = (7)

^{2}
⇒ 6/n = 2

Since, the bases are same and equating the powers, we get n =

~~6~~/

~~2~~ = 3.

`

● **Exponents**

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**Laws of Exponents**

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**Integral Exponents of a Rational Numbers**

**Solved Examples on Exponents**

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