Precalculus Examples

${(x^{2} - 5)}^{\frac{3}{2}} = 27$

Step 1

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

${({(x^{2} - 5)}^{\frac{3}{2}})}^{\frac{2}{3}} = 27^{\frac{2}{3}}$

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 ${({(x^{2} - 5)}^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 2.1.1.1

Multiply the exponents in ${({(x^{2} - 5)}^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 2.1.1.1.1

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

${(x^{2} - 5)}^{\frac{3}{2} \cdot \frac{2}{3}} = 27^{\frac{2}{3}}$

Step 2.1.1.1.2

Cancel the common factor of $3$ .

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Step 2.1.1.1.2.1

Cancel the common factor.

${(x^{2} - 5)}^{\frac{3}{2} \cdot \frac{2}{3}} = 27^{\frac{2}{3}}$

Step 2.1.1.1.2.2

Rewrite the expression.

${(x^{2} - 5)}^{\frac{1}{2} \cdot 2} = 27^{\frac{2}{3}}$

Step 2.1.1.1.3

Cancel the common factor of $2$ .

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Step 2.1.1.1.3.1

Cancel the common factor.

${(x^{2} - 5)}^{\frac{1}{2} \cdot 2} = 27^{\frac{2}{3}}$

Step 2.1.1.1.3.2

Rewrite the expression.

${(x^{2} - 5)}^{1} = 27^{\frac{2}{3}}$

Step 2.1.1.2

Simplify.

$x^{2} - 5 = 27^{\frac{2}{3}}$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $27^{\frac{2}{3}}$ .

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Step 2.2.1.1

Simplify the expression.

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Step 2.2.1.1.1

Rewrite $27$ as $3^{3}$ .

$x^{2} - 5 = {(3^{3})}^{\frac{2}{3}}$

Step 2.2.1.1.2

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

$x^{2} - 5 = 3^{3 (\frac{2}{3})}$

Step 2.2.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.2.1

Cancel the common factor.

$x^{2} - 5 = 3^{3 (\frac{2}{3})}$

Step 2.2.1.2.2

Rewrite the expression.

$x^{2} - 5 = 3^{2}$

Step 2.2.1.3

Raise $3$ to the power of $2$ .

$x^{2} - 5 = 9$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Add $5$ to both sides of the equation.

$x^{2} = 9 + 5$

Step 3.1.2

Add $9$ and $5$ .

$x^{2} = 14$

Step 3.2

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

$x = \pm \sqrt{14}$

Step 3.3

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

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Step 3.3.1

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

$x = \sqrt{14}$

Step 3.3.2

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

$x = - \sqrt{14}$

Step 3.3.3

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

$x = \sqrt{14}, - \sqrt{14}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{14}, - \sqrt{14}$

Decimal Form:

$x = 3.74165738 \dots, - 3.74165738 \dots$