Basic Math Examples

$\sqrt{y^{2} - 16^{2}} = 5$

Step 1

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

${\sqrt{y^{2} - 16^{2}}}^{2} = 5^{2}$

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 $\sqrt{y^{2} - 16^{2}}$ as ${(y^{2} - 16^{2})}^{\frac{1}{2}}$ .

${({(y^{2} - 16^{2})}^{\frac{1}{2}})}^{2} = 5^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(y^{2} - 16^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(y^{2} - 16^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(y^{2} - 16^{2})}^{\frac{1}{2} \cdot 2} = 5^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(y^{2} - 16^{2})}^{\frac{1}{2} \cdot 2} = 5^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(y^{2} - 16^{2})}^{1} = 5^{2}$

Step 2.2.1.2

Simplify each term.

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Step 2.2.1.2.1

Raise $16$ to the power of $2$ .

${(y^{2} - 1 \cdot 256)}^{1} = 5^{2}$

Step 2.2.1.2.2

Multiply $- 1$ by $256$ .

${(y^{2} - 256)}^{1} = 5^{2}$

Step 2.2.1.3

Simplify.

$y^{2} - 256 = 5^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

$y^{2} - 256 = 25$

Step 3

Solve for $y$ .

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Step 3.1

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

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Step 3.1.1

Add $256$ to both sides of the equation.

$y^{2} = 25 + 256$

Step 3.1.2

Add $25$ and $256$ .

$y^{2} = 281$

Step 3.2

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

$y = \pm \sqrt{281}$

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.

$y = \sqrt{281}$

Step 3.3.2

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

$y = - \sqrt{281}$

Step 3.3.3

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

$y = \sqrt{281}, - \sqrt{281}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$y = \sqrt{281}, - \sqrt{281}$

Decimal Form:

$y = 16.76305461 \dots, - 16.76305461 \dots$