Basic Math Examples

$\sqrt{y^{2} - 16} = 344$

Step 1

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

${\sqrt{y^{2} - 16}}^{2} = 344^{2}$

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

${({(y^{2} - 16)}^{\frac{1}{2}})}^{2} = 344^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

${(y^{2} - 16)}^{\frac{1}{2} \cdot 2} = 344^{2}$

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.

${(y^{2} - 16)}^{\frac{1}{2} \cdot 2} = 344^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(y^{2} - 16)}^{1} = 344^{2}$

Step 2.2.1.2

Simplify.

$y^{2} - 16 = 344^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Raise $344$ to the power of $2$ .

$y^{2} - 16 = 118336$

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 $16$ to both sides of the equation.

$y^{2} = 118336 + 16$

Step 3.1.2

Add $118336$ and $16$ .

$y^{2} = 118352$

Step 3.2

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

$y = \pm \sqrt{118352}$

Step 3.3

Simplify $\pm \sqrt{118352}$ .

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Step 3.3.1

Rewrite $118352$ as $4^{2} \cdot 7397$ .

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Step 3.3.1.1

Factor $16$ out of $118352$ .

$y = \pm \sqrt{16 (7397)}$

Step 3.3.1.2

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

$y = \pm \sqrt{4^{2} \cdot 7397}$

Step 3.3.2

Pull terms out from under the radical.

$y = \pm 4 \sqrt{7397}$

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.

$y = 4 \sqrt{7397}$

Step 3.4.2

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

$y = - 4 \sqrt{7397}$

Step 3.4.3

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

$y = 4 \sqrt{7397}, - 4 \sqrt{7397}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$y = 4 \sqrt{7397}, - 4 \sqrt{7397}$

Decimal Form:

$y = 344.02325502 \dots, - 344.02325502 \dots$