Algebra Examples

$\sqrt{{(y - 4)}^{2}} = 5$

Step 1

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

${\sqrt{{(y - 4)}^{2}}}^{2} = 5^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{{(y - 4)}^{2}}$ as ${(y - 4)}^{\frac{2}{2}}$ .

${({(y - 4)}^{\frac{2}{2}})}^{2} = 5^{2}$

Step 2.2

Divide $2$ by $2$ .

${({(y - 4)}^{1})}^{2} = 5^{2}$

Step 2.3

Simplify the left side.

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Step 2.3.1

Multiply the exponents in ${({(y - 4)}^{1})}^{2}$ .

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Step 2.3.1.1

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

${(y - 4)}^{1 \cdot 2} = 5^{2}$

Step 2.3.1.2

Multiply $2$ by $1$ .

${(y - 4)}^{2} = 5^{2}$

Step 2.4

Simplify the right side.

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Step 2.4.1

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

${(y - 4)}^{2} = 25$

Step 3

Solve for $y$ .

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Step 3.1

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

$y - 4 = \pm \sqrt{25}$

Step 3.2

Simplify $\pm \sqrt{25}$ .

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Step 3.2.1

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

$y - 4 = \pm \sqrt{5^{2}}$

Step 3.2.2

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

$y - 4 = \pm 5$

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.

$y - 4 = 5$

Step 3.3.2

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

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Step 3.3.2.1

Add $4$ to both sides of the equation.

$y = 5 + 4$

Step 3.3.2.2

Add $5$ and $4$ .

$y = 9$

Step 3.3.3

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

$y - 4 = - 5$

Step 3.3.4

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

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Step 3.3.4.1

Add $4$ to both sides of the equation.

$y = - 5 + 4$

Step 3.3.4.2

Add $- 5$ and $4$ .

$y = - 1$

Step 3.3.5

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

$y = 9, - 1$