Exponents — Multiple Choice Questions

$\left(-\dfrac{2}{3}\right)^{-3}$ is equal to :

1. $\dfrac{27}{8}$

2. $\dfrac{-8}{27}$

3. $\dfrac{-27}{8}$

4. $\dfrac{8}{27}$

Answer

Solving,

$\Rightarrow \left(-\dfrac{2}{3}\right)^{-3}\\[1em] = \left(-\dfrac{3}{2}\right)^{3}\\[1em] = \dfrac{(-3) \times (-3) \times (-3)}{2 \times 2 \times 2}\\[1em] = \dfrac{-27}{8}$

Hence, Option 3 is the correct option.

(-3)² ÷ $\left(-\dfrac{1}{2}\right)^3$ is equal to :

1. $-\dfrac{9}{8}$

2. $\dfrac{9}{8}$

3. -72

4. 72

Answer

Solving,

$\Rightarrow (-3)^2 ÷ \left(-\dfrac{1}{2}\right)^3\\[1em] = 9 ÷ \left(-\dfrac{1}{8}\right)\\[1em] = 9 \times (-8)\\[1em] = -72$

Hence, Option 3 is the correct option.

The reciprocal of (-2)⁵ is :

1. -32

2. $-\dfrac{1}{32}$

3. 32

4. $\dfrac{1}{32}$

Answer

Solving,

⇒ (-2)⁵ = (-2) × (-2) × (-2) × (-2) × (-2) = -32.

Reciprocal of -32 = $\dfrac{1}{-32} = -\dfrac{1}{32}$

Hence, Option 2 is the correct option.

If $\left(\dfrac{1}{5}\right)^3 \times \left(\dfrac{1}{5}\right)^{x+3}$ = 5^-2, then x is equal to :

1. 4

2. $\dfrac{1}{4}$

3. $-\dfrac{1}{4}$

4. -4

Answer

Solving,

$\Rightarrow \left(\dfrac{1}{5}\right)^3 \times \left(\dfrac{1}{5}\right)^{x+3} = 5^{-2}\\[1em] \Rightarrow \left(\dfrac{1}{5}\right)^{3 + x + 3} = 5^{-2}\\[1em] \Rightarrow \left(\dfrac{1}{5}\right)^{x + 6} = 5^{-2}\\[1em] \Rightarrow (5^{-1})^{x + 6} = 5^{-2}\\[1em] \Rightarrow 5^{-(x + 6)} = 5^{-2}$

Comparing the powers of the same base,

⇒ -(x + 6) = -2

⇒ x + 6 = 2

⇒ x = -4

Hence, Option 4 is the correct option.

4⁰ + 6⁰ - 8⁰ is equal to :

1. 0

2. 1

3. 2

4. $-\dfrac{1}{2}$

Answer

As any non-zero base raised to the power zero is equal to 1,

⇒ 4⁰ + 6⁰ - 8⁰

⇒ 1 + 1 - 1

⇒ 1.

Hence, Option 2 is the correct option.

(2³)² ÷ (2²)⁴ is equal to :

1. 4

2. -4

3. $-\dfrac{1}{4}$

4. $\dfrac{1}{4}$

Answer

Solving,

⇒ (2³)² ÷ (2²)⁴

= 2^{3 × 2} ÷ 2^{2 × 4}

= 2⁶ ÷ 2⁸

= 2^6-8

= 2^-2

= $\dfrac{1}{2^2} = \dfrac{1}{4}$

Hence, Option 4 is the correct option.

If $\left(-\dfrac{2}{3}\right)^x = -\dfrac{243}{32}$, then x is equal to :

1. -5

2. 5

3. $\dfrac{1}{5}$

4. $-\dfrac{1}{5}$

Answer

Solving,

$\Rightarrow \left(-\dfrac{2}{3}\right)^x = -\dfrac{243}{32}\\[1em] \Rightarrow \left(-\dfrac{2}{3}\right)^x = -\dfrac{3^5}{2^5}\\[1em] \Rightarrow \left(-\dfrac{2}{3}\right)^x = \left(-\dfrac{3}{2}\right)^5\\[1em] \Rightarrow \left(-\dfrac{2}{3}\right)^x = \left(-\dfrac{2}{3}\right)^{-5}$

Comparing the powers of the same base,

⇒ x = -5.

Hence, Option 1 is the correct option.

(-4)³ ÷ (4)⁴ is equal to :

1. 4

2. -4

3. $-\dfrac{1}{4}$

4. $\dfrac{1}{4}$

Answer

Solving,

$\Rightarrow (-4)^3 ÷ (4)^4\\[1em] = \dfrac{(-4)^3}{4^4}\\[1em] = \dfrac{-64}{256}\\[1em] = -\dfrac{1}{4}$

Hence, Option 3 is the correct option.

7⁰ × 5 - (-2)³ - 8⁰ is equal to :

1. 12

2. -12

3. $\dfrac{1}{12}$

4. $-\dfrac{1}{12}$

Answer

Solving,

⇒ 7⁰ × 5 - (-2)³ - 8⁰

⇒ (1 × 5) - (-8) - 1

⇒ 5 + 8 - 1

⇒ 12.

Hence, Option 1 is the correct option.

If $\left(\dfrac{9}{10}\right)^2 \times \left(\dfrac{10}{9}\right)^5 = \left(\dfrac{9}{10}\right)^{1-m}$, then m is equal to :

1. 4

2. -4

3. $\dfrac{1}{4}$

4. $-\dfrac{1}{4}$

Answer

Solving,

$\Rightarrow \left(\dfrac{9}{10}\right)^2 \times \left(\dfrac{10}{9}\right)^5 = \left(\dfrac{9}{10}\right)^{1-m}\\[1em] \Rightarrow \left(\dfrac{9}{10}\right)^2 \times \left(\dfrac{9}{10}\right)^{-5} = \left(\dfrac{9}{10}\right)^{1-m}\\[1em] \Rightarrow \left(\dfrac{9}{10}\right)^{2 - 5} = \left(\dfrac{9}{10}\right)^{1-m}\\[1em] \Rightarrow \left(\dfrac{9}{10}\right)^{-3} = \left(\dfrac{9}{10}\right)^{1-m}$

Comparing the powers of the same base,

⇒ 1 - m = -3

⇒ m = 4

Hence, Option 1 is the correct option.

$\left(\dfrac{3}{5}\right)^{-1} \div \left(\dfrac{-5}{2}\right)^{-1}$ is equal to :

1. $\dfrac{25}{6}$

2. $-\dfrac{25}{6}$

3. $\dfrac{6}{25}$

4. $-\dfrac{6}{25}$

Answer

Solving,

$\Rightarrow \left(\dfrac{3}{5}\right)^{-1} \div \left(\dfrac{-5}{2}\right)^{-1}\\[1em] = \dfrac{5}{3} \div \left(-\dfrac{2}{5}\right)\\[1em] = \dfrac{5}{3} \times \left(-\dfrac{5}{2}\right)\\[1em] = -\dfrac{25}{6}$

Hence, Option 2 is the correct option.

If (-3)^{x - 1} = - 243, then (243)^x-6 is equal to :

1. (243)^-12

2. 243

3. 0

4. 1

Answer

Solving,

⇒ (-3)^{x - 1} = -243

⇒ (-3)^{x - 1} = (-3)⁵

Comparing the powers of the same base,

⇒ x - 1 = 5

⇒ x = 6

Now,

⇒ (243)^x-6 = (243)^{6 - 6} = (243)⁰ = 1.

Hence, Option 4 is the correct option.

{(5²)³ × 5⁴} ÷ 5³ is equal to :

1. 5¹³

2. 5⁷

3. 5²¹

4. none of these

Answer

⇒ {(5²)³ × 5⁴} ÷ 5³

⇒ {5^{2 × 3} × 5⁴} ÷ 5³

⇒ {5⁶ × 5⁴} ÷ 5³

⇒ 5^{6 + 4} ÷ 5³

⇒ 5¹⁰ ÷ 5³

⇒ 5^{10 - 3}

⇒ 5⁷.

Hence, Option 2 is the correct option.

If (3^-1) × x = (6)^-1, then the value of x is :

1. $\dfrac{1}{2}$

2. $-\dfrac{1}{2}$

3. 2

4. -2

Answer

Solving,

$\Rightarrow (3^{-1}) \times x = (6)^{-1}\\[1em] \Rightarrow \dfrac{1}{3} \times x = \dfrac{1}{6}\\[1em] \Rightarrow x = \dfrac{1}{6} \times 3\\[1em] \Rightarrow x = \dfrac{3}{6}\\[1em] \Rightarrow x = \dfrac{1}{2}$

Hence, Option 1 is the correct option.

If (-9)^-1 ÷ x = (18)^-1, then the value of x is :

1. 2

2. -2

3. $\dfrac{1}{2}$

4. $-\dfrac{1}{2}$

Answer

Solving,

$\Rightarrow (-9)^{-1} ÷ x = (18)^{-1}\\[1em] \Rightarrow -\dfrac{1}{9} ÷ x = \dfrac{1}{18}\\[1em] \Rightarrow -\dfrac{1}{9} \times \dfrac{1}{x} = \dfrac{1}{18}\\[1em] \Rightarrow \dfrac{1}{x} = \dfrac{1}{18} \times (-9)\\[1em] \Rightarrow \dfrac{1}{x} = -\dfrac{9}{18}\\[1em] \Rightarrow \dfrac{1}{x} = -\dfrac{1}{2}\\[1em] \Rightarrow x = -2$

Hence, Option 2 is the correct option.