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 = 1/5³ = 1/125

(ii) (1/3)^-4 = (3/1)⁴ = 3⁴ = 81



(iii) (5/2)^-3 = (2/5)³ = 2³/5³ = 8/125

(iv) (-2)^-5 = 1/(-2)^-5 = 1/-2⁵ = 1/-32 = -1/32

(v) (-3/4)^-4 = (4/-3)⁴ = (-4/3)⁴ = (-4)⁴/3⁴ = 4⁴/3⁴ = 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)⁴/2⁴ × (-5)²/7²

= {7⁴ × (-5)²}/{2⁴ × 7² } [Since, (-7)⁴ = 7⁴]

= {7² × (-5)² }/2⁴

= [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

= -4⁵

= -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)⁶/3⁶

= 2⁶/3⁶

= 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

= (2² + 3² + 4²)

= (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)³/3³ = 4²/9² × 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} = 3¹¹ ÷ 3⁵

Solution:

3^{2n + 3} = 3¹¹ ÷ 3⁵/3^-7

⇒ 3^{2n + 3} = 3^{11 - 5}/3^-7

⇒ 3^{2n + 3} = 3⁶/3^-7

⇒ 3^{2n + 3} = 3^{6 - (-7)}

⇒ 3^{2n + 3} = 3^{6 + 7}

⇒ 3^{2n + 3} = 3¹³

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 3ⁿ = 243

Solution:

3ⁿ = 3⁵

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) = 3ⁿ

⇒ (3)³ = 3ⁿ

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.

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Integral Exponents of a Rational Numbers

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