Precalculus Examples

${(x^{2} + 9)}^{\frac{1}{2}} = 5$

Step 1

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${({(x^{2} + 9)}^{\frac{1}{2}})}^{2} = 5^{2}$

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 ${({(x^{2} + 9)}^{\frac{1}{2}})}^{2}$ .

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Step 2.1.1.1

Multiply the exponents in ${({(x^{2} + 9)}^{\frac{1}{2}})}^{2}$ .

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Step 2.1.1.1.1

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

${(x^{2} + 9)}^{\frac{1}{2} \cdot 2} = 5^{2}$

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.

${(x^{2} + 9)}^{\frac{1}{2} \cdot 2} = 5^{2}$

Step 2.1.1.1.2.2

Rewrite the expression.

${(x^{2} + 9)}^{1} = 5^{2}$

Step 2.1.1.2

Simplify.

$x^{2} + 9 = 5^{2}$

Step 2.2

Simplify the right side.

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Step 2.2.1

Raise $5$ to the power of $2$ .

$x^{2} + 9 = 25$

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

Subtract $9$ from both sides of the equation.

$x^{2} = 25 - 9$

Step 3.1.2

Subtract $9$ from $25$ .

$x^{2} = 16$

Step 3.2

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

$x = \pm \sqrt{16}$

Step 3.3

Simplify $\pm \sqrt{16}$ .

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Step 3.3.1

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

$x = \pm \sqrt{4^{2}}$

Step 3.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 4$

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.

$x = 4$

Step 3.4.2

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

$x = - 4$

Step 3.4.3

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

$x = 4, - 4$