Precalculus Examples

$\sqrt{x^{2} + 15} - 4 = 0$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 15}$ as ${(x^{2} + 15)}^{\frac{1}{2}}$ .

${(x^{2} + 15)}^{\frac{1}{2}} - 4 = 0$

Step 2

Add $4$ to both sides of the equation.

${(x^{2} + 15)}^{\frac{1}{2}} = 4$

Step 3

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

${({(x^{2} + 15)}^{\frac{1}{2}})}^{2} = 4^{2}$

Step 4

Simplify the exponent.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify ${({(x^{2} + 15)}^{\frac{1}{2}})}^{2}$ .

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Step 4.1.1.1

Multiply the exponents in ${({(x^{2} + 15)}^{\frac{1}{2}})}^{2}$ .

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Step 4.1.1.1.1

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

${(x^{2} + 15)}^{\frac{1}{2} \cdot 2} = 4^{2}$

Step 4.1.1.1.2

Cancel the common factor of $2$ .

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Step 4.1.1.1.2.1

Cancel the common factor.

${(x^{2} + 15)}^{\frac{1}{2} \cdot 2} = 4^{2}$

Step 4.1.1.1.2.2

Rewrite the expression.

${(x^{2} + 15)}^{1} = 4^{2}$

Step 4.1.1.2

Simplify.

$x^{2} + 15 = 4^{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Raise $4$ to the power of $2$ .

$x^{2} + 15 = 16$

Step 5

Solve for $x$ .

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Step 5.1

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

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Step 5.1.1

Subtract $15$ from both sides of the equation.

$x^{2} = 16 - 15$

Step 5.1.2

Subtract $15$ from $16$ .

$x^{2} = 1$

Step 5.2

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

$x = \pm \sqrt{1}$

Step 5.3

Any root of $1$ is $1$ .

$x = \pm 1$

Step 5.4

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

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Step 5.4.1

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

$x = 1$

Step 5.4.2

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

$x = - 1$

Step 5.4.3

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

$x = 1, - 1$