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Exponents and Surds

Exponential Equations

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Quiz

Exponential Equations Score: 0/10

1. Solve for x : 3 ( 9^{x + 3} ) = 27^{2x - 1}

A. x = 10

B. x =

\frac{2}{5}

C. x =

\frac{5}{2}

D. None of the above

2. Solve for x : 2^x = 0.125

A. x = -3

B. x = 3

C. x = 8

D. None of the above.

3. Solve for x : 5^x - 30 = - 5^{x + 1}

A. x = 0

B. x = 1

C. x = 5

D. None of the above.

4. Solve for x : 5^{x + 2} - 5^x = 23 ( 5^x ) + 1

A. x = 0

B. x = 1

C. x = 5

D. None of the above.

5. Solve for x : 5^x + 15 ( 5^{- x} ) = 2

A. x = 5

B. x = 1

C. x = - 3

D. None of the above.

6. Solve for x : x^2/3 = x^1/3 + 12

A. x = - 64 or x = 27

B. x = 64 or x = 27

C. x = 64 or x = - 27

D. None of the above.

7. Solve for x : x^{- 3/2} = 8

A. x =

\frac{1}{4}

B. x = 4

C. x = 1

D. None of the above.

8. Solve for x : $\sqrt[5]{x 4}$ = 256

A. x = 1024

B. x = - 1024

C. x = 1024 or x = - 1024

D. None of the above.

9. Solve for x : $\sqrt[]{3x + 4}$ - 5 = 0

A. x = 7

B. x = 0

C. x = 7 or x = 0

D. None of the above.

10. Solve for x : $\sqrt[]{3x - 5}$ - 5 = x

A. x = 3 or x = 10

B. x = 10

C. x = 3

D. None of the above.

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Quiz Results

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Solutions

Exponential Equations Question 1 of 10

1. Solve for x : 3 ( 9^{x + 3} ) = 27^{2x - 1}

A. x = 10

B. x =

\frac{2}{5}

C. x =

\frac{5}{2}

D. None of the above

Correct Answer
Option C

Detailed Solution

Solution Details

Given 3 ( 9^{x + 3} ) = 27^{2x - 1}

• 3 ( 9^{x + 3} ) = 27^{2x - 1}

3 ( ( 3² ) ^{x + 3} ) = ( 3³ ) ^{2x - 1}

3 ( 3^{2x + 6} ) = 3^{6x - 3}

3^{1 + 2x + 6} = 3^{6x - 3}

2x + 7 = 6x - 3

- 4x = - 10

∴ x = $\frac{5}{2}$

For more info please see

Exponential Equations

Exponential Equations Question 2 of 10

2. Solve for x : 2^x = 0.125

A. x = -3

B. x = 3

C. x = 8

D. None of the above.

Correct Answer
Option A

Detailed Solution

Solution Details

Given 2^x = 0.125

• 2^x = 0.125

2^x = $\frac{125}{1000}$ = $\frac{1}{8}$ = $\frac{1}{23}$

2^x = 2 ^{- 3}

∴ x = - 3

For more info please see

Exponential Equations

Exponential Equations Question 3 of 10

3. Solve for x : 5^x - 30 = - 5^{x + 1}

A. x = 0

B. x = 1

C. x = 5

D. None of the above.

Correct Answer
Option B

Detailed Solution

Solution Details

Given 5^x - 30 = 5^{x + 1}

• 5^x - 30 = - 5^{x + 1}

5^x + 5^{x + 1} = 30

5^x + 5^x5¹ = 30

5^x ( 1 + 5 ) = 30

5^x ( 6 ) = 30

5^x = 5

∴ x = 1

For more info please see

Exponential Equations

Exponential Equations Question 4 of 10

4. Solve for x : 5^{x + 2} - 5^x = 23 ( 5^x ) + 1

A. x = 0

B. x = 1

C. x = 5

D. None of the above.

Correct Answer
Option A

Detailed Solution

Solution Details

Given 5^{x + 2} - 5^x = 23 ( 5^x ) + 1

• 5^{x + 2} - 5^x = 23 ( 5^x ) + 1

5^{x + 2} - 5^x - 23 ( 5^x ) = 1

5^{x + 2} - 24 ( 5^x ) = 1

5^x 5² - 24 ( 5^x ) = 1

5^x ( 5² - 24 ) = 1

5^x ( 1 ) = 1

5^x = 5⁰

∴ x = 0

For more info please see

Exponential Equations

Exponential Equations Question 5 of 10

5. Solve for x : 5^x + 15 ( 5^{- x} ) = 2

A. x = 5

B. x = 1

C. x = - 3

D. None of the above.

Correct Answer
Option B

Detailed Solution

Solution Details

Given 5^x + 15 ( 5^{- x} ) = 2

• 5^x + 15 ( 5^{- x} ) = 2

5^x + 15 ( 5^{- x} ) = 2

5^x + $\frac{15}{5 x}$ = 2

5^x ( 5^x ) + $\frac{15}{5 x}$ ( 5^x ) = 2 ( 5^x )

5^2x + 15 = 2 ( 5^x )

( 5^x )² - 2 ( 5^x ) + 15 = 0

( 5^x - 5 ) ( 5^x + 3 ) = 0

( 5^x - 5 ) = 0 or ( 5^x + 3 ) = 0

For ( 5^x + 3 ) There is no solution

5^x = 5

∴ x = 1

For more info please see

Exponential Equations

Exponential Equations Question 6 of 10

6. Solve for x : x^2/3 = x^1/3 + 12

A. x = - 64 or x = 27

B. x = 64 or x = 27

C. x = 64 or x = - 27

D. None of the above.

Correct Answer
Option C

Detailed Solution

Solution Details

Given x^2/3 = x^1/3 + 12

• x^2/3 = x^1/3 + 12

( x^1/3 )² - x^1/3 - 12 = 0

( x^1/3 - 4) ( x^1/3 + 3 ) = 0

( x^1/3 - 4) = 0 or ( x^1/3 + 3 ) = 0

∴ x = 64 or x = - 27

For more info please see

Exponential Equations

Exponential Equations Question 7 of 10

7. Solve for x : x^{- 3/2} = 8

A. x =

\frac{1}{4}

B. x = 4

C. x = 1

D. None of the above.

Correct Answer
Option A

Detailed Solution

Solution Details

Given x^{- 3/2} = 8

• x^{- 3/2} = 8

( x^{- 3/2} )^{- 2/3} = ( 2³ )^{- 2/3}

x = 2^{- 2}

∴ x = $\frac{1}{4}$

For more info please see

Exponential Equations

Exponential Equations Question 8 of 10

8. Solve for x : $\sqrt[5]{x 4}$ = 256

A. x = 1024

B. x = - 1024

C. x = 1024 or x = - 1024

D. None of the above.

Correct Answer
Option C

Detailed Solution

Solution Details

Given $\sqrt[5]{x 4}$ = 256

• $\sqrt[5]{x 4}$ = 256

x^4/5 = 2⁸ or x^4/5 = - 2⁸

( x^4/5 )^5/4 = ( 2⁸ )^5/4 or ( x^4/5 )^5/4 = (- 2⁸ )^5/4

∴ x = 1024 or x = - 1024

For more info please see

Exponential Equations

Given $\sqrt[5]{x 4}$ = 256

• $\sqrt[5]{x 4}$ = 256

x^4/5 = 2⁸ or x^4/5 = - 2⁸

( x^4/5 )^5/4 = ( 2⁸ )^5/4 or

( x^4/5 )^5/4 = (- 2⁸ )^5/4

x = 1024 or x = - 1024

For more info please see

Exponential Equations

Exponential Equations Question 9 of 10

9. Solve for x : $\sqrt[]{3x + 4}$ - 5 = 0

A. x = 7

B. x = 0

C. x = 7 or x = 0

D. None of the above.

Correct Answer
Option A

Detailed Solution

Solution Details

Given $\sqrt[]{3x + 4}$ - 5 = 0

• $\sqrt[]{3x + 4}$ - 5 = 0

$\sqrt[]{3x + 4}$ = 5

( $\sqrt[]{3x + 4}$ ) ² = ( 5 )²

3x + 4 = 25

3x = 21

x = 7

Check if the value of x is a solution to the equation:

For x = 7

$\sqrt[]{3x + 4}$ - 5 = 0

$\sqrt[]{3(7) + 4}$ - 5 = 0

LHS = 0 and RHS = 0

∴ x = 7 is a solution.

For more info please see

Exponential Equations

Exponential Equations Question 10 of 10

10. Solve for x : $\sqrt[]{3x - 5}$ - 5 = x

A. x = 3 or x = 10

B. x = 10

C. x = 3

D. None of the above.

Correct Answer
Option B

Detailed Solution

Solution Details

Given $\sqrt[]{3x - 5}$ + 5 = x

• $\sqrt[]{3x - 5}$ + 5 = x

$\sqrt[]{3x - 5}$ = x - 5

( $\sqrt[]{3x - 5}$ )² = ( x - 5 )²

3x - 5 = x² - 10x + 25

x²- 13x + 30 = 0

( x - 10 ) (x - 3 ) = 0

( x - 10 ) = 0 or (x - 3 ) = 0

x = 10 or x = 3

Check if the values of x are solutions to the equation:

For x = 3

$\sqrt[]{3x - 5}$ + 5 = x

$\sqrt[]{3(3) - 5}$ + 5 = 7

LHS = 7 and RHS = 3

∴ x = 3 is not a solution.

For x = 10

$\sqrt[]{3x - 5}$ + 5 = x

$\sqrt[]{3(10) - 5}$ + 5 = 10

LHS = 10 and RHS = 10

∴ x = 10 is the only solution.

For more info please see

Exponential Equations

Quiz

Exponents and Surds

Exponential Equations

Quiz Quidelines

Quiz

1. Solve for x : 3 ( 9 x + 3 ) = 27 2x - 1

2. Solve for x : 2 x = 0.125

3. Solve for x : 5 x - 30 = - 5 x + 1

4. Solve for x : 5 x + 2 - 5 x = 23 ( 5 x ) + 1

5. Solve for x : 5 x + 15 ( 5 - x ) = 2

6. Solve for x : x 2/3 = x 1/3 + 12

7. Solve for x : x - 3/2 = 8

8. Solve for x : x 4 5 = 256

9. Solve for x : 3x + 4 - 5 = 0

10. Solve for x : 3x - 5 - 5 = x

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Quiz Results

Solutions

1. Solve for x : 3 ( 9 x + 3 ) = 27 2x - 1

Correct Answer Option C

Detailed Solution

Solution Details

Given 3 ( 9 x + 3 ) = 27 2x - 1

• 3 ( 9 x + 3 ) = 27 2x - 1 3 ( ( 3 2 ) x + 3 ) = ( 3 3 ) 2x - 1 3 ( 3 2x + 6 ) = 3 6x - 3 3 1 + 2x + 6 = 3 6x - 3 2x + 7 = 6x - 3 - 4x = - 10 ∴ x = 5 2

2. Solve for x : 2 x = 0.125

Correct Answer Option A

Detailed Solution

Solution Details

Given 2 x = 0.125

• 2 x = 0.125 2 x = 125 1000 = 1 8 = 1 2 3 2 x = 2 - 3 ∴ x = - 3

3. Solve for x : 5 x - 30 = - 5 x + 1

Correct Answer Option B

Detailed Solution

Solution Details

Given 5 x - 30 = 5 x + 1

• 5 x - 30 = - 5 x + 1 5 x + 5 x + 1 = 30 5 x + 5 x 5 1 = 30 5 x ( 1 + 5 ) = 30 5 x ( 6 ) = 30 5 x = 5 ∴ x = 1

4. Solve for x : 5 x + 2 - 5 x = 23 ( 5 x ) + 1

Correct Answer Option A

Detailed Solution

Solution Details

Given 5 x + 2 - 5 x = 23 ( 5 x ) + 1

• 5 x + 2 - 5 x = 23 ( 5 x ) + 1 5 x + 2 - 5 x - 23 ( 5 x ) = 1 5 x + 2 - 24 ( 5 x ) = 1 5 x 5 2 - 24 ( 5 x ) = 1 5 x ( 5 2 - 24 ) = 1 5 x ( 1 ) = 1 5 x = 5 0 ∴ x = 0

5. Solve for x : 5 x + 15 ( 5 - x ) = 2

Correct Answer Option B

Detailed Solution

Solution Details

Given 5 x + 15 ( 5 - x ) = 2

• 5 x + 15 ( 5 - x ) = 2 5 x + 15 ( 5 - x ) = 2 5 x + 15 5 x = 2 5 x ( 5 x ) + 15 5 x ( 5 x ) = 2 ( 5 x ) 5 2x + 15 = 2 ( 5 x ) ( 5 x ) 2 - 2 ( 5 x ) + 15 = 0 ( 5 x - 5 ) ( 5 x + 3 ) = 0 ( 5 x - 5 ) = 0 or ( 5 x + 3 ) = 0 For ( 5 x + 3 ) There is no solution 5 x = 5 ∴ x = 1

6. Solve for x : x 2/3 = x 1/3 + 12

Correct Answer Option C

Detailed Solution

Solution Details

Given x 2/3 = x 1/3 + 12

• x 2/3 = x 1/3 + 12 ( x 1/3 ) 2 - x 1/3 - 12 = 0 ( x 1/3 - 4) ( x 1/3 + 3 ) = 0 ( x 1/3 - 4) = 0 or ( x 1/3 + 3 ) = 0 ∴ x = 64 or x = - 27

7. Solve for x : x - 3/2 = 8

Correct Answer Option A

Detailed Solution

Solution Details

Given x - 3/2 = 8

• x - 3/2 = 8 ( x - 3/2 ) - 2/3 = ( 2 3 ) - 2/3 x = 2 - 2 ∴ x = 1 4

8. Solve for x : x 4 5 = 256

Correct Answer Option C

Detailed Solution

Solution Details

Given x 4 5 = 256

• x 4 5 = 256 x 4/5 = 2 8 or x 4/5 = - 2 8 ( x 4/5 ) 5/4 = ( 2 8 ) 5/4 or ( x 4/5 ) 5/4 = (- 2 8 ) 5/4 ∴ x = 1024 or x = - 1024

Given x 4 5 = 256

• x 4 5 = 256 x 4/5 = 2 8 or x 4/5 = - 2 8 ( x 4/5 ) 5/4 = ( 2 8 ) 5/4 or ( x 4/5 ) 5/4 = (- 2 8 ) 5/4 x = 1024 or x = - 1024

9. Solve for x : 3x + 4 - 5 = 0

Correct Answer Option A

Detailed Solution

Solution Details

Given 3x + 4 - 5 = 0

• 3x + 4 - 5 = 0 3x + 4 = 5 ( 3x + 4 ) 2 = ( 5 ) 2 3x + 4 = 25 3x = 21 x = 7

Check if the value of x is a solution to the equation: For x = 7 3x + 4 - 5 = 0 3(7) + 4 - 5 = 0

LHS = 0 and RHS = 0 ∴ x = 7 is a solution.

10. Solve for x : 3x - 5 - 5 = x

Correct Answer Option B

Detailed Solution

Solution Details

1. Solve for x : 3 ( 9^{x + 3} ) = 27^{2x - 1}

2. Solve for x : 2^x = 0.125

3. Solve for x : 5^x - 30 = - 5^{x + 1}

4. Solve for x : 5^{x + 2} - 5^x = 23 ( 5^x ) + 1

5. Solve for x : 5^x + 15 ( 5^{- x} ) = 2

6. Solve for x : x^2/3 = x^1/3 + 12

7. Solve for x : x^{- 3/2} = 8

8. Solve for x : $\sqrt[5]{x 4}$ = 256

9. Solve for x : $\sqrt[]{3x + 4}$ - 5 = 0

10. Solve for x : $\sqrt[]{3x - 5}$ - 5 = x

1. Solve for x : 3 ( 9^{x + 3} ) = 27^{2x - 1}

Correct Answer
Option C

Given 3 ( 9^{x + 3} ) = 27^{2x - 1}

• 3 ( 9^{x + 3} ) = 27^{2x - 1}

3 ( ( 3² ) ^{x + 3} ) = ( 3³ ) ^{2x - 1}

3 ( 3^{2x + 6} ) = 3^{6x - 3}

3^{1 + 2x + 6} = 3^{6x - 3}

2x + 7 = 6x - 3

- 4x = - 10

∴ x = $\frac{5}{2}$

2. Solve for x : 2^x = 0.125

Correct Answer
Option A

Given 2^x = 0.125

• 2^x = 0.125

2^x = $\frac{125}{1000}$ = $\frac{1}{8}$ = $\frac{1}{23}$

2^x = 2 ^{- 3}

∴ x = - 3

3. Solve for x : 5^x - 30 = - 5^{x + 1}

Correct Answer
Option B

Given 5^x - 30 = 5^{x + 1}

• 5^x - 30 = - 5^{x + 1}

5^x + 5^{x + 1} = 30

5^x + 5^x5¹ = 30

5^x ( 1 + 5 ) = 30

5^x ( 6 ) = 30

5^x = 5

∴ x = 1

4. Solve for x : 5^{x + 2} - 5^x = 23 ( 5^x ) + 1

Correct Answer
Option A

Given 5^{x + 2} - 5^x = 23 ( 5^x ) + 1

• 5^{x + 2} - 5^x = 23 ( 5^x ) + 1

5^{x + 2} - 5^x - 23 ( 5^x ) = 1

5^{x + 2} - 24 ( 5^x ) = 1

5^x 5² - 24 ( 5^x ) = 1

5^x ( 5² - 24 ) = 1

5^x ( 1 ) = 1

5^x = 5⁰

∴ x = 0

5. Solve for x : 5^x + 15 ( 5^{- x} ) = 2

Correct Answer
Option B

Given 5^x + 15 ( 5^{- x} ) = 2

6. Solve for x : x^2/3 = x^1/3 + 12

Correct Answer
Option C

Given x^2/3 = x^1/3 + 12

• x^2/3 = x^1/3 + 12

( x^1/3 )² - x^1/3 - 12 = 0

( x^1/3 - 4) ( x^1/3 + 3 ) = 0

( x^1/3 - 4) = 0 or ( x^1/3 + 3 ) = 0

∴ x = 64 or x = - 27

7. Solve for x : x^{- 3/2} = 8

Correct Answer
Option A

Given x^{- 3/2} = 8

• x^{- 3/2} = 8

( x^{- 3/2} )^{- 2/3} = ( 2³ )^{- 2/3}

x = 2^{- 2}

∴ x = $\frac{1}{4}$

8. Solve for x : $\sqrt[5]{x 4}$ = 256

Correct Answer
Option C

Given $\sqrt[5]{x 4}$ = 256

• $\sqrt[5]{x 4}$ = 256

x^4/5 = 2⁸ or x^4/5 = - 2⁸

( x^4/5 )^5/4 = ( 2⁸ )^5/4 or ( x^4/5 )^5/4 = (- 2⁸ )^5/4

∴ x = 1024 or x = - 1024

Given $\sqrt[5]{x 4}$ = 256

• $\sqrt[5]{x 4}$ = 256

x^4/5 = 2⁸ or x^4/5 = - 2⁸

( x^4/5 )^5/4 = ( 2⁸ )^5/4 or

( x^4/5 )^5/4 = (- 2⁸ )^5/4

x = 1024 or x = - 1024

9. Solve for x : $\sqrt[]{3x + 4}$ - 5 = 0

Correct Answer
Option A

Given $\sqrt[]{3x + 4}$ - 5 = 0

• $\sqrt[]{3x + 4}$ - 5 = 0

$\sqrt[]{3x + 4}$ = 5

( $\sqrt[]{3x + 4}$ ) ² = ( 5 )²

3x + 4 = 25

3x = 21

x = 7

Check if the value of x is a solution to the equation:

For x = 7

$\sqrt[]{3x + 4}$ - 5 = 0

$\sqrt[]{3(7) + 4}$ - 5 = 0

LHS = 0 and RHS = 0

∴ x = 7 is a solution.

10. Solve for x : $\sqrt[]{3x - 5}$ - 5 = x

Correct Answer
Option B

Given $\sqrt[]{3x - 5}$ + 5 = x

• $\sqrt[]{3x - 5}$ + 5 = x

$\sqrt[]{3x - 5}$ = x - 5

( $\sqrt[]{3x - 5}$ )² = ( x - 5 )²

3x - 5 = x² - 10x + 25

x²- 13x + 30 = 0

( x - 10 ) (x - 3 ) = 0

( x - 10 ) = 0 or (x - 3 ) = 0

x = 10 or x = 3