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Exponents and Surds

Typical Exam Questions

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Exponential Equations Score: 0/10

1. Simplify and choose the correct answer: $\frac{6 6x 9 3x}{54 4x 4 x - 2}$

1. Simplify and choose the correct answer:

$\frac{6 6x 9 3x}{54 4x 4 x - 2}$

\frac{1}{16}

B. 1

C. 16

D. None of the above.

2. Simplify and choose the correct answer: $\frac{2 2x + 2 - 2 2x - 1}{4 x + 8 ( 4 x - 2)}$

2. Simplify and choose the correct answer:

$\frac{2 2x + 2 - 2 2x - 1}{4 x + 8 ( 4 x - 2)}$

\frac{3}{7}

\frac{7}{3}

C. 1

D.None of the above.

3. Solve for x : 3^x = $\sqrt[3]{\frac{1}{9}}$

A. x =

\frac{3}{2}

B. x = 1

C. x = -

\frac{2}{3}

D. None of the above.

4. Solve for x : 2^{x + 2} = $\sqrt[]{32}$

A. x =

\frac{5}{2}

B. x = 2

C. x =

\frac{1}{2}

D. None of the above.

5. Solve for x : x ( x ^1/2 + x ^{- 1/2} ) ² = 0

A. x = 0 or x = - 1

B. x = 0

C. x = - 1

D. None of the above.

6. Solve for x : $\sqrt[]{x}$ - x ^{- 1} = 0

A. x = 1

B. x = 1 or x = 0

C. x = 0

D. None of the above.

7. Solve for x : $\sqrt[]{2}$ ( x + $\sqrt[]{2}$ ) = x + $\sqrt[]{2}$

A. x = 2

B. x = -

\sqrt[]{2}

C. x =

\sqrt[]{2}

D. None of the above.

8. Solve for x : 3^x 2^{x + 1} = 72

A. x = 2

B. x = 0

C. x = - 2

D. None of the above.

9. Solve for x : 4^{( x + 1 ) ( x - 3 )} = 8^{- x}

A. x = 2

B. x = -

\frac{3}{2}

or x = 2

C. x = -

\frac{3}{2}

D. None of the above.

10. Solve for x : 2^2x - 3 ( 2^x ) - 4 = 0

A. x = 2 or x = 0

B. x = 2

C. x = 0

D. None of the above.

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Quiz Results

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Solutions

Exponential Equations Question 1 of 10

1. Simplify and choose the correct answer: $\frac{6 6x 9 3x}{54 4x 4 x - 2}$

1. Simplify and choose the correct answer:

$\frac{6 6x 9 3x}{54 4x 4 x - 2}$

\frac{1}{16}

B. 1

C. 16

D. None of the above.

Correct Answer
Option C

Detailed Solution

Solution Details

Given $\frac{6 6x 9 3x}{54 4x 4 x - 2}$

• $\frac{6 6x 9 3x}{54 4x 4 x - 2}$

= $\frac{( 2 x 3 ) 6x ( 3 2) 3x}{( 2 x 3 3) 4x ( 2 2) x - 2}$

= $\frac{2 6x 3 6x 3 6x}{2 4x 3 12x 2 2x - 4}$

= 2^{6x - 4x - 2x + 4} 3^{6x + 6x - 12x}

= 2⁴ 3⁰

= 16

For more info please see

Exponential Equations

Exponential Equations Question 2 of 10

2. Simplify and choose the correct answer: $\frac{2 2x + 2 - 2 2x - 1}{4 x + 8 ( 4 x - 2)}$

2. Simplify and choose the correct answer:

$\frac{2 2x + 2 - 2 2x - 1}{4 x + 8 ( 4 x - 2)}$

\frac{3}{7}

\frac{7}{3}

C. 1

D. None of the above.

Correct Answer
Option B

Detailed Solution

Solution Details

Given $\frac{2 2x + 2 - 2 2x - 1}{4 x + 8 ( 4 x - 2)}$

• $\frac{2 2x + 2 - 2 2x - 1}{4 x + 8 ( 4 x - 2)}$

= $\frac{2 2x + 2 - 2 2x - 1}{( 2 2) x + 2 3 ( 2 2) x - 2}$

= $\frac{2 2x 22 - 2 2x 2 - 1}{2 2x + 2 32 2x 2 - 4}$

= $\frac{2 2x ( 2 2 - 2 - 1)}{2 2x ( 1 + 2 32 - 4)}$

= $\frac{22 - 2 - 1}{1 + 2 - 1}$

= $\frac{4 - \frac{1}{2}}{1 + \frac{1}{2}}$

= $\frac{\frac{7}{2}}{\frac{3}{2}}$

= $\frac{7}{2}$ x $\frac{2}{3}$

= $\frac{7}{3}$

For more info please see

Exponential Equations

Exponential Equations Question 3 of 10

3. Solve for x : 3^x = $\sqrt[3]{\frac{1}{9}}$

A. x =

\frac{3}{2}

B. x = 1

C. x = -

\frac{2}{3}

D. None of the above.

Correct Answer
Option C

Detailed Solution

Solution Details

Given 3^x = $\sqrt[3]{\frac{1}{9}}$

• 3^x = $\sqrt[3]{\frac{1}{9}}$

3^x = $\frac{\sqrt[3]{1}}{\sqrt[3]{9}}$

3^x = $\frac{1}{\sqrt[3]{32}}$

3^x = $\frac{1}{3 2/3}$

3^x ( 3 ^2/3 ) = 1

3^x 3 ^2/3 = 3⁰

x + $\frac{2}{3}$ = 0

x = - $\frac{2}{3}$

For more info please see

Exponential Equations

Exponential Equations Question 4 of 10

4. Solve for x : 2^{x + 2} = $\sqrt[]{32}$

A. x =

\frac{5}{2}

B. x = 2

C. x =

\frac{1}{2}

D. None of the above.

Correct Answer
Option C

Detailed Solution

Solution Details

Given 2^{x + 2} = $\sqrt[]{32}$

• 2^{x + 2} = $\sqrt[]{32}$

2^{x + 2} = $\sqrt[]{16 x 2}$

2^{x + 2} = 4 $\sqrt[]{2}$

2^{x + 2} = 2² 2^1/2

x + 2 = 2 + $\frac{1}{2}$

x = $\frac{5}{2}$ - 2

x = $\frac{1}{2}$

For more info please see

Exponential Equations

Exponential Equations Question 5 of 10

5. Solve for x : x ( x ^1/2 - x ^{- 1/2} ) ² = 0

A. x = 0 or x = - 1

B. x = 0

C. x = - 1

D. None of the above.

Correct Answer
Option C

Detailed Solution

Solution Details

Given x ( x ^1/2 + x ^{- 1/2} ) ² = 0

• x ( x ^1/2 + x ^{- 1/2} ) ² = 0

x ≠ 0 or ( x ^1/2 + x ^{- 1/2} ) ² = 0

( x ^1/2 + x ^{- 1/2} ) ² = 0

x ^1/2 + x ^{- 1/2} = 0

x ^1/2 ( x ^1/2 ) + x ^{- 1/2} ( x ^1/2 ) = 0

x + 1 = 0

x = - 1

For more info please see

Exponential Equations

Exponential Equations Question 6 of 10

6. Solve for x : $\sqrt[]{x}$ - x ^{- 1} = 0

A. x = 1

B. x = 1 or x = 0

C. x = 0

D. None of the above.

Correct Answer
Option A

Detailed Solution

Solution Details

Given $\sqrt[]{x}$ - x ^{- 1} = 0

• $\sqrt[]{x}$ - x ^{- 1} = 0

x ^1/2 = x ^{- 1}

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

x ^3/2 = 1

( x ^3/2 )^2/3 = 1^2/3

∴ x = 1

For more info please see

Exponential Equations

Exponential Equations Question 7 of 10

7. Solve for x : $\sqrt[]{2}$ ( x + $\sqrt[]{2}$ ) = x + $\sqrt[]{2}$

A. x = 2

B. x = -

\sqrt[]{2}

C. x =

\sqrt[]{2}

D. None of the above.

Correct Answer
Option B

Detailed Solution

Solution Details

Given $\sqrt[]{2}$ ( x + $\sqrt[]{2}$ ) = x + $\sqrt[]{2}$

• $\sqrt[]{2}$ ( x + $\sqrt[]{2}$ ) = x + $\sqrt[]{2}$

$\sqrt[]{2}$ x + 2 = x + $\sqrt[]{2}$

$\sqrt[]{2}$ x - x = $\sqrt[]{2}$ - 2

x ( $\sqrt[]{2}$ - 1 ) = $\sqrt[]{2}$ - 2

x = $\frac{\sqrt[]{2} - 2}{\sqrt[]{2} - 1}$

x = $\frac{(\sqrt[]{2} - 2) (\sqrt[]{2} + 1)}{(\sqrt[]{2} - 1) (\sqrt[]{2} + 1)}$

∴ x = - $\sqrt[]{2}$

For more info please see

Exponential Equations

Exponential Equations Question 8 of 10

8. Solve for x : 3^x 2^{x + 1} = 72

A. x = 2

B. x = 0

C. x = - 2

D. None of the above.

Correct Answer
Option A

Detailed Solution

Solution Details

Given 3^x 2^{x + 1} = 72

• 3^x 2^{x + 1} = 72

3^x 2^x 2¹ = 72

( 3 . 2 )^x = 36

6^x = 6²

x = 2

For more info please see

Exponential Equations

Exponential Equations Question 9 of 10

9. Solve for x : 4^{( x + 1 ) ( x - 3 )} = 8^{- x}

A. x = 2

B. x = -

\frac{3}{2}

or x = 2

C. x = -

\frac{3}{2}

D. None of the above.

Correct Answer
Option B

Detailed Solution

Solution Details

Given 4^{( x + 1 ) ( x - 3 )} = 8^{- x}

• 4^{( x + 1 ) ( x - 3 )} = 8^{- x}

2 ²^{( x + 1 ) ( x - 3 )} = 2 ^{- 3x}

2 ( x + 1 ) ( x - 3 ) = - 3x

2x² - 4x - 6 = - 3x

2x² - x - 6 = 0

( 2x + 3 ) ( x - 2 ) = 0

x = - $\frac{3}{2}$ or x = 2

For more info please see

Exponential Equations

Exponential Equations Question 10 of 10

10. Solve for x : 2^2x - 3 ( 2^x ) - 4 = 0

A. x = 2 or x = 0

B. x = 2

C. x = 0

D. None of the above.

Correct Answer
Option B

Detailed Solution

Solution Details

Given 2^2x - 3 ( 2^x ) - 4 = 0

• 2^2x - 3 ( 2^x ) - 4 = 0

( 2^x + 1 ) ( 2^x - 4 ) = 0

For ( 2^x + 1 ) = 0 there is no solution

So ( 2^x - 4 ) = 0

2^x = 2²

x = 2

For more info please see

Exponential Equations

Quiz

Exponents and Surds

Typical Exam Questions

Quiz Quidelines

Quiz

1. Simplify and choose the correct answer: 6 6x 9 3x 54 4x 4 x - 2

1. Simplify and choose the correct answer: 6 6x 9 3x 54 4x 4 x - 2

1. Simplify and choose the correct answer: 6 6x 9 3x 54 4x 4 x - 2

2. Simplify and choose the correct answer: 2 2x + 2 - 2 2x - 1 4 x + 8 ( 4 x - 2 )

2. Simplify and choose the correct answer: 2 2x + 2 - 2 2x - 1 4 x + 8 ( 4 x - 2 )

2. Simplify and choose the correct answer: 2 2x + 2 - 2 2x - 1 4 x + 8 ( 4 x - 2 )

3. Solve for x : 3 x = 1 9 3

4. Solve for x : 2 x + 2 = 32

5. Solve for x : x ( x 1/2 + x - 1/2 ) 2 = 0

6. Solve for x : x - x - 1 = 0

7. Solve for x : 2 ( x + 2 ) = x + 2

8. Solve for x : 3 x 2 x + 1 = 72

9. Solve for x : 4 ( x + 1 ) ( x - 3 ) = 8 - x

10. Solve for x : 2 2x - 3 ( 2 x ) - 4 = 0

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Quiz Results

Solutions

1. Simplify and choose the correct answer: 6 6x 9 3x 54 4x 4 x - 2

1. Simplify and choose the correct answer: 6 6x 9 3x 54 4x 4 x - 2

1. Simplify and choose the correct answer: 6 6x 9 3x 54 4x 4 x - 2

Correct Answer Option C

Detailed Solution

Solution Details

Given 6 6x 9 3x 54 4x 4 x - 2

• 6 6x 9 3x 54 4x 4 x - 2 = ( 2 x 3 ) 6x ( 3 2 ) 3x ( 2 x 3 3 ) 4x ( 2 2 ) x - 2 = 2 6x 3 6x 3 6x 2 4x 3 12x 2 2x - 4 = 2 6x - 4x - 2x + 4 3 6x + 6x - 12x = 2 4 3 0 = 16

2. Simplify and choose the correct answer: 2 2x + 2 - 2 2x - 1 4 x + 8 ( 4 x - 2 )

2. Simplify and choose the correct answer: 2 2x + 2 - 2 2x - 1 4 x + 8 ( 4 x - 2 )

2. Simplify and choose the correct answer: 2 2x + 2 - 2 2x - 1 4 x + 8 ( 4 x - 2 )

Correct Answer Option B

Detailed Solution

Solution Details

Given 2 2x + 2 - 2 2x - 1 4 x + 8 ( 4 x - 2 )

• 2 2x + 2 - 2 2x - 1 4 x + 8 ( 4 x - 2 ) = 2 2x + 2 - 2 2x - 1 ( 2 2 ) x + 2 3 ( 2 2 ) x - 2 = 2 2x 2 2 - 2 2x 2 - 1 2 2x + 2 3 2 2x 2 - 4 = 2 2x ( 2 2 - 2 - 1 ) 2 2x ( 1 + 2 3 2 - 4 ) = 2 2 - 2 - 1 1 + 2 - 1 = 4 - 1 2 1 + 1 2 = 7 2 3 2 = 7 2 x 2 3 = 7 3

3. Solve for x : 3 x = 1 9 3

Correct Answer Option C

Detailed Solution

Solution Details

Given 3 x = 1 9 3

• 3 x = 1 9 3 3 x = 1 3 9 3 3 x = 1 3 2 3 3 x = 1 3 2/3 3 x ( 3 2/3 ) = 1 3 x 3 2/3 = 3 0 x + 2 3 = 0 x = - 2 3

4. Solve for x : 2 x + 2 = 32

Correct Answer Option C

Detailed Solution

Solution Details

Given 2 x + 2 = 32

• 2 x + 2 = 32 2 x + 2 = 16 x 2 2 x + 2 = 4 2 2 x + 2 = 2 2 2 1/2 x + 2 = 2 + 1 2 x = 5 2 - 2 x = 1 2

5. Solve for x : x ( x 1/2 - x - 1/2 ) 2 = 0

Correct Answer Option C

Detailed Solution

Solution Details

Given x ( x 1/2 + x - 1/2 ) 2 = 0

• x ( x 1/2 + x - 1/2 ) 2 = 0 x ≠ 0 or ( x 1/2 + x - 1/2 ) 2 = 0 ( x 1/2 + x - 1/2 ) 2 = 0 x 1/2 + x - 1/2 = 0 x 1/2 ( x 1/2 ) + x - 1/2 ( x 1/2 ) = 0 x + 1 = 0 x = - 1

6. Solve for x : x - x - 1 = 0

Correct Answer Option A

Detailed Solution

Solution Details

Given x - x - 1 = 0

• x - x - 1 = 0 x 1/2 = x - 1 x 1/2 ( x ) = x - 1 ( x ) x 3/2 = 1 ( x 3/2 ) 2/3 = 1 2/3 ∴ x = 1

7. Solve for x : 2 ( x + 2 ) = x + 2

Correct Answer Option B

Detailed Solution

Solution Details

Given 2 ( x + 2 ) = x + 2

• 2 ( x + 2 ) = x + 2 2 x + 2 = x + 2 2 x - x = 2 - 2 x ( 2 - 1 ) = 2 - 2 x = 2 - 2 2 - 1 x = ( 2 - 2 ) ( 2 + 1) ( 2 - 1) ( 2 + 1) ∴ x = - 2

8. Solve for x : 3 x 2 x + 1 = 72

Correct Answer Option A

Detailed Solution

Solution Details

Given 3 x 2 x + 1 = 72

• 3 x 2 x + 1 = 72 3 x 2 x 2 1 = 72 ( 3 . 2 ) x = 36 6 x = 6 2 x = 2

9. Solve for x : 4 ( x + 1 ) ( x - 3 ) = 8 - x

Correct Answer Option B

Detailed Solution

Solution Details

Given 4 ( x + 1 ) ( x - 3 ) = 8 - x

• 4 ( x + 1 ) ( x - 3 ) = 8 - x 2 2 ( x + 1 ) ( x - 3 ) = 2 - 3x 2 ( x + 1 ) ( x - 3 ) = - 3x 2x 2 - 4x - 6 = - 3x 2x 2 - x - 6 = 0 ( 2x + 3 ) ( x - 2 ) = 0 x = - 3 2 or x = 2

1. Simplify and choose the correct answer: $\frac{6 6x 9 3x}{54 4x 4 x - 2}$

1. Simplify and choose the correct answer: $\frac{6 6x 9 3x}{54 4x 4 x - 2}$

1. Simplify and choose the correct answer:

$\frac{6 6x 9 3x}{54 4x 4 x - 2}$

2. Simplify and choose the correct answer: $\frac{2 2x + 2 - 2 2x - 1}{4 x + 8 ( 4 x - 2)}$

2. Simplify and choose the correct answer: $\frac{2 2x + 2 - 2 2x - 1}{4 x + 8 ( 4 x - 2)}$

2. Simplify and choose the correct answer:

$\frac{2 2x + 2 - 2 2x - 1}{4 x + 8 ( 4 x - 2)}$

3. Solve for x : 3^x = $\sqrt[3]{\frac{1}{9}}$

4. Solve for x : 2^{x + 2} = $\sqrt[]{32}$

5. Solve for x : x ( x ^1/2 + x ^{- 1/2} ) ² = 0

6. Solve for x : $\sqrt[]{x}$ - x ^{- 1} = 0

7. Solve for x : $\sqrt[]{2}$ ( x + $\sqrt[]{2}$ ) = x + $\sqrt[]{2}$

8. Solve for x : 3^x 2^{x + 1} = 72

9. Solve for x : 4^{( x + 1 ) ( x - 3 )} = 8^{- x}

10. Solve for x : 2^2x - 3 ( 2^x ) - 4 = 0

1. Simplify and choose the correct answer: $\frac{6 6x 9 3x}{54 4x 4 x - 2}$

1. Simplify and choose the correct answer: $\frac{6 6x 9 3x}{54 4x 4 x - 2}$

1. Simplify and choose the correct answer:

$\frac{6 6x 9 3x}{54 4x 4 x - 2}$

Correct Answer
Option C

Given $\frac{6 6x 9 3x}{54 4x 4 x - 2}$

• $\frac{6 6x 9 3x}{54 4x 4 x - 2}$

= $\frac{( 2 x 3 ) 6x ( 3 2) 3x}{( 2 x 3 3) 4x ( 2 2) x - 2}$

= $\frac{2 6x 3 6x 3 6x}{2 4x 3 12x 2 2x - 4}$

= 2^{6x - 4x - 2x + 4} 3^{6x + 6x - 12x}

= 2⁴ 3⁰

= 16

2. Simplify and choose the correct answer: $\frac{2 2x + 2 - 2 2x - 1}{4 x + 8 ( 4 x - 2)}$

2. Simplify and choose the correct answer: $\frac{2 2x + 2 - 2 2x - 1}{4 x + 8 ( 4 x - 2)}$

2. Simplify and choose the correct answer:

$\frac{2 2x + 2 - 2 2x - 1}{4 x + 8 ( 4 x - 2)}$

Correct Answer
Option B

Given $\frac{2 2x + 2 - 2 2x - 1}{4 x + 8 ( 4 x - 2)}$

3. Solve for x : 3^x = $\sqrt[3]{\frac{1}{9}}$

Correct Answer
Option C

Given 3^x = $\sqrt[3]{\frac{1}{9}}$

• 3^x = $\sqrt[3]{\frac{1}{9}}$

3^x = $\frac{\sqrt[3]{1}}{\sqrt[3]{9}}$

3^x = $\frac{1}{\sqrt[3]{32}}$

3^x = $\frac{1}{3 2/3}$

3^x ( 3 ^2/3 ) = 1

3^x 3 ^2/3 = 3⁰

x + $\frac{2}{3}$ = 0

x = - $\frac{2}{3}$

4. Solve for x : 2^{x + 2} = $\sqrt[]{32}$

Correct Answer
Option C

Given 2^{x + 2} = $\sqrt[]{32}$

• 2^{x + 2} = $\sqrt[]{32}$

2^{x + 2} = $\sqrt[]{16 x 2}$

2^{x + 2} = 4 $\sqrt[]{2}$

2^{x + 2} = 2² 2^1/2

x + 2 = 2 + $\frac{1}{2}$

x = $\frac{5}{2}$ - 2

x = $\frac{1}{2}$

5. Solve for x : x ( x ^1/2 - x ^{- 1/2} ) ² = 0

Correct Answer
Option C

Given x ( x ^1/2 + x ^{- 1/2} ) ² = 0

• x ( x ^1/2 + x ^{- 1/2} ) ² = 0

x ≠ 0 or ( x ^1/2 + x ^{- 1/2} ) ² = 0

( x ^1/2 + x ^{- 1/2} ) ² = 0

x ^1/2 + x ^{- 1/2} = 0

x ^1/2 ( x ^1/2 ) + x ^{- 1/2} ( x ^1/2 ) = 0

x + 1 = 0

x = - 1

6. Solve for x : $\sqrt[]{x}$ - x ^{- 1} = 0

Correct Answer
Option A

Given $\sqrt[]{x}$ - x ^{- 1} = 0

• $\sqrt[]{x}$ - x ^{- 1} = 0

x ^1/2 = x ^{- 1}

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

x ^3/2 = 1

( x ^3/2 )^2/3 = 1^2/3

∴ x = 1

7. Solve for x : $\sqrt[]{2}$ ( x + $\sqrt[]{2}$ ) = x + $\sqrt[]{2}$

Correct Answer
Option B

Given $\sqrt[]{2}$ ( x + $\sqrt[]{2}$ ) = x + $\sqrt[]{2}$

8. Solve for x : 3^x 2^{x + 1} = 72

Correct Answer
Option A

Given 3^x 2^{x + 1} = 72

• 3^x 2^{x + 1} = 72

3^x 2^x 2¹ = 72

( 3 . 2 )^x = 36

6^x = 6²

x = 2

9. Solve for x : 4^{( x + 1 ) ( x - 3 )} = 8^{- x}

Correct Answer
Option B

Given 4^{( x + 1 ) ( x - 3 )} = 8^{- x}

• 4^{( x + 1 ) ( x - 3 )} = 8^{- x}

2 ²^{( x + 1 ) ( x - 3 )} = 2 ^{- 3x}

2 ( x + 1 ) ( x - 3 ) = - 3x

2x² - 4x - 6 = - 3x

2x² - x - 6 = 0

( 2x + 3 ) ( x - 2 ) = 0

x = - $\frac{3}{2}$ or x = 2

10. Solve for x : 2^2x - 3 ( 2^x ) - 4 = 0

Correct Answer
Option B

Given 2^2x - 3 ( 2^x ) - 4 = 0

• 2^2x - 3 ( 2^x ) - 4 = 0

( 2^x + 1 ) ( 2^x - 4 ) = 0

For ( 2^x + 1 ) = 0 there is no solution

So ( 2^x - 4 ) = 0

2^x = 2²

x = 2