Basic Math Examples

$\sqrt{76 - 2 c} = \sqrt{\frac{c}{9}}$

Step 1

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

${\sqrt{76 - 2 c}}^{2} = {\sqrt{\frac{c}{9}}}^{2}$

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 $\sqrt{76 - 2 c}$ as ${(76 - 2 c)}^{\frac{1}{2}}$ .

${({(76 - 2 c)}^{\frac{1}{2}})}^{2} = {\sqrt{\frac{c}{9}}}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(76 - 2 c)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(76 - 2 c)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(76 - 2 c)}^{\frac{1}{2} \cdot 2} = {\sqrt{\frac{c}{9}}}^{2}$

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.

${(76 - 2 c)}^{\frac{1}{2} \cdot 2} = {\sqrt{\frac{c}{9}}}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(76 - 2 c)}^{1} = {\sqrt{\frac{c}{9}}}^{2}$

Step 2.2.1.2

Simplify.

$76 - 2 c = {\sqrt{\frac{c}{9}}}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${\sqrt{\frac{c}{9}}}^{2}$ .

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Step 2.3.1.1

Rewrite $\frac{c}{9}$ as ${(\frac{1}{3})}^{2} c$ .

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Step 2.3.1.1.1

Factor the perfect power $1^{2}$ out of $c$ .

$76 - 2 c = {\sqrt{\frac{1^{2} c}{9}}}^{2}$

Step 2.3.1.1.2

Factor the perfect power $3^{2}$ out of $9$ .

$76 - 2 c = {\sqrt{\frac{1^{2} c}{3^{2} \cdot 1}}}^{2}$

Step 2.3.1.1.3

Rearrange the fraction $\frac{1^{2} c}{3^{2} \cdot 1}$ .

$76 - 2 c = {\sqrt{{(\frac{1}{3})}^{2} c}}^{2}$

Step 2.3.1.2

Pull terms out from under the radical.

$76 - 2 c = {(\frac{1}{3} \sqrt{c})}^{2}$

Step 2.3.1.3

Combine $\frac{1}{3}$ and $\sqrt{c}$ .

$76 - 2 c = {(\frac{\sqrt{c}}{3})}^{2}$

Step 2.3.1.4

Apply the product rule to $\frac{\sqrt{c}}{3}$ .

$76 - 2 c = \frac{{\sqrt{c}}^{2}}{3^{2}}$

Step 2.3.1.5

Rewrite ${\sqrt{c}}^{2}$ as $c$ .

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Step 2.3.1.5.1

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

$76 - 2 c = \frac{{(c^{\frac{1}{2}})}^{2}}{3^{2}}$

Step 2.3.1.5.2

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

$76 - 2 c = \frac{c^{\frac{1}{2} \cdot 2}}{3^{2}}$

Step 2.3.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$76 - 2 c = \frac{c^{\frac{2}{2}}}{3^{2}}$

Step 2.3.1.5.4

Cancel the common factor of $2$ .

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Step 2.3.1.5.4.1

Cancel the common factor.

$76 - 2 c = \frac{c^{\frac{2}{2}}}{3^{2}}$

Step 2.3.1.5.4.2

Rewrite the expression.

$76 - 2 c = \frac{c^{1}}{3^{2}}$

Step 2.3.1.5.5

Simplify.

$76 - 2 c = \frac{c}{3^{2}}$

Step 2.3.1.6

Raise $3$ to the power of $2$ .

$76 - 2 c = \frac{c}{9}$

Step 3

Solve for $c$ .

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Step 3.1

Multiply both sides by $9$ .

$(76 - 2 c) \cdot 9 = \frac{c}{9} \cdot 9$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $(76 - 2 c) \cdot 9$ .

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Step 3.2.1.1.1

Apply the distributive property.

$76 \cdot 9 - 2 c \cdot 9 = \frac{c}{9} \cdot 9$

Step 3.2.1.1.2

Simplify the expression.

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Step 3.2.1.1.2.1

Multiply $76$ by $9$ .

$684 - 2 c \cdot 9 = \frac{c}{9} \cdot 9$

Step 3.2.1.1.2.2

Multiply $9$ by $- 2$ .

$684 - 18 c = \frac{c}{9} \cdot 9$

Step 3.2.1.1.2.3

Reorder $684$ and $- 18 c$ .

$- 18 c + 684 = \frac{c}{9} \cdot 9$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Cancel the common factor of $9$ .

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Step 3.2.2.1.1

Cancel the common factor.

$- 18 c + 684 = \frac{c}{9} \cdot 9$

Step 3.2.2.1.2

Rewrite the expression.

$- 18 c + 684 = c$

Step 3.3

Solve for $c$ .

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Step 3.3.1

Move all terms containing $c$ to the left side of the equation.

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Step 3.3.1.1

Subtract $c$ from both sides of the equation.

$- 18 c + 684 - c = 0$

Step 3.3.1.2

Subtract $c$ from $- 18 c$ .

$- 19 c + 684 = 0$

Step 3.3.2

Subtract $684$ from both sides of the equation.

$- 19 c = - 684$

Step 3.3.3

Divide each term in $- 19 c = - 684$ by $- 19$ and simplify.

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Step 3.3.3.1

Divide each term in $- 19 c = - 684$ by $- 19$ .

$\frac{- 19 c}{- 19} = \frac{- 684}{- 19}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 19$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 19 c}{- 19} = \frac{- 684}{- 19}$

Step 3.3.3.2.1.2

Divide $c$ by $1$ .

$c = \frac{- 684}{- 19}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Divide $- 684$ by $- 19$ .

$c = 36$