Precalculus Examples

$13 = \sqrt{12^{2} + y^{2}}$

Step 1

Rewrite the equation as $\sqrt{12^{2} + y^{2}} = 13$ .

$\sqrt{12^{2} + y^{2}} = 13$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{12^{2} + y^{2}}}^{2} = 13^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{12^{2} + y^{2}}$ as ${(12^{2} + y^{2})}^{\frac{1}{2}}$ .

${({(12^{2} + y^{2})}^{\frac{1}{2}})}^{2} = 13^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(12^{2} + y^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(12^{2} + y^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(12^{2} + y^{2})}^{\frac{1}{2} \cdot 2} = 13^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(12^{2} + y^{2})}^{\frac{1}{2} \cdot 2} = 13^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(12^{2} + y^{2})}^{1} = 13^{2}$

Step 3.2.1.2

Raise $12$ to the power of $2$ .

${(144 + y^{2})}^{1} = 13^{2}$

Step 3.2.1.3

Simplify.

$144 + y^{2} = 13^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Raise $13$ to the power of $2$ .

$144 + y^{2} = 169$

Step 4

Solve for $y$ .

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Step 4.1

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

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Step 4.1.1

Subtract $144$ from both sides of the equation.

$y^{2} = 169 - 144$

Step 4.1.2

Subtract $144$ from $169$ .

$y^{2} = 25$

Step 4.2

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

$y = \pm \sqrt{25}$

Step 4.3

Simplify $\pm \sqrt{25}$ .

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Step 4.3.1

Rewrite $25$ as $5^{2}$ .

$y = \pm \sqrt{5^{2}}$

Step 4.3.2

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

$y = \pm 5$

Step 4.4

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

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Step 4.4.1

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

$y = 5$

Step 4.4.2

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

$y = - 5$

Step 4.4.3

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

$y = 5, - 5$