Precalculus Examples

${(x^{2} + 6)}^{\frac{2}{3}} = 16$

Step 1

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

${({(x^{2} + 6)}^{\frac{2}{3}})}^{\frac{3}{2}} = \pm 16^{\frac{3}{2}}$

Step 2

Simplify the exponent.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify ${({(x^{2} + 6)}^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 2.1.1.1

Multiply the exponents in ${({(x^{2} + 6)}^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 2.1.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x^{2} + 6)}^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 16^{\frac{3}{2}}$

Step 2.1.1.1.2

Cancel the common factor of $2$ .

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Step 2.1.1.1.2.1

Cancel the common factor.

${(x^{2} + 6)}^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 16^{\frac{3}{2}}$

Step 2.1.1.1.2.2

Rewrite the expression.

${(x^{2} + 6)}^{\frac{1}{3} \cdot 3} = \pm 16^{\frac{3}{2}}$

Step 2.1.1.1.3

Cancel the common factor of $3$ .

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Step 2.1.1.1.3.1

Cancel the common factor.

${(x^{2} + 6)}^{\frac{1}{3} \cdot 3} = \pm 16^{\frac{3}{2}}$

Step 2.1.1.1.3.2

Rewrite the expression.

${(x^{2} + 6)}^{1} = \pm 16^{\frac{3}{2}}$

Step 2.1.1.2

Simplify.

$x^{2} + 6 = \pm 16^{\frac{3}{2}}$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $\pm 16^{\frac{3}{2}}$ .

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Step 2.2.1.1

Simplify the expression.

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Step 2.2.1.1.1

Rewrite $16$ as $4^{2}$ .

$x^{2} + 6 = \pm {(4^{2})}^{\frac{3}{2}}$

Step 2.2.1.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{2} + 6 = \pm 4^{2 (\frac{3}{2})}$

Step 2.2.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.2.1

Cancel the common factor.

$x^{2} + 6 = \pm 4^{2 (\frac{3}{2})}$

Step 2.2.1.2.2

Rewrite the expression.

$x^{2} + 6 = \pm 4^{3}$

Step 2.2.1.3

Raise $4$ to the power of $3$ .

$x^{2} + 6 = \pm 64$

Step 3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.1

First, use the positive value of the $\pm$ to find the first solution.

$x^{2} + 6 = 64$

Step 3.2

Move all terms not containing $x$ to the right side of the equation.

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Step 3.2.1

Subtract $6$ from both sides of the equation.

$x^{2} = 64 - 6$

Step 3.2.2

Subtract $6$ from $64$ .

$x^{2} = 58$

Step 3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{58}$

Step 3.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{58}$

Step 3.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{58}$

Step 3.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{58}, - \sqrt{58}$

Step 3.5

Next, use the negative value of the $\pm$ to find the second solution.

$x^{2} + 6 = - 64$

Step 3.6

Move all terms not containing $x$ to the right side of the equation.

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Step 3.6.1

Subtract $6$ from both sides of the equation.

$x^{2} = - 64 - 6$

Step 3.6.2

Subtract $6$ from $- 64$ .

$x^{2} = - 70$

Step 3.7

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 70}$

Step 3.8

Simplify $\pm \sqrt{- 70}$ .

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Step 3.8.1

Rewrite $- 70$ as $- 1 (70)$ .

$x = \pm \sqrt{- 1 (70)}$

Step 3.8.2

Rewrite $\sqrt{- 1 (70)}$ as $\sqrt{- 1} \cdot \sqrt{70}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{70}$

Step 3.8.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \sqrt{70}$

Step 3.9

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.9.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i \sqrt{70}$

Step 3.9.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i \sqrt{70}$

Step 3.9.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i \sqrt{70}, - i \sqrt{70}$

Step 3.10

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{58}, - \sqrt{58}, i \sqrt{70}, - i \sqrt{70}$