Basic Math Examples

$684 = \frac{76 p r^{3}}{3}$

Step 1

Rewrite the equation as $\frac{76 p r^{3}}{3} = 684$ .

$\frac{76 p r^{3}}{3} = 684$

Step 2

Multiply both sides of the equation by $\frac{3}{76}$ .

$\frac{3}{76} \cdot \frac{76 p r^{3}}{3} = \frac{3}{76} \cdot 684$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{3}{76} \cdot \frac{76 p r^{3}}{3}$ .

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Step 3.1.1.1

Combine.

$\frac{3 (76 p r^{3})}{76 \cdot 3} = \frac{3}{76} \cdot 684$

Step 3.1.1.2

Cancel the common factor of $3$ .

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Step 3.1.1.2.1

Cancel the common factor.

$\frac{3 (76 p r^{3})}{76 \cdot 3} = \frac{3}{76} \cdot 684$

Step 3.1.1.2.2

Rewrite the expression.

$\frac{76 p r^{3}}{76} = \frac{3}{76} \cdot 684$

Step 3.1.1.3

Cancel the common factor of $76$ .

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Step 3.1.1.3.1

Cancel the common factor.

$\frac{76 p r^{3}}{76} = \frac{3}{76} \cdot 684$

Step 3.1.1.3.2

Divide $p r^{3}$ by $1$ .

$p r^{3} = \frac{3}{76} \cdot 684$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\frac{3}{76} \cdot 684$ .

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Step 3.2.1.1

Cancel the common factor of $76$ .

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Step 3.2.1.1.1

Factor $76$ out of $684$ .

$p r^{3} = \frac{3}{76} \cdot (76 (9))$

Step 3.2.1.1.2

Cancel the common factor.

$p r^{3} = \frac{3}{76} \cdot (76 \cdot 9)$

Step 3.2.1.1.3

Rewrite the expression.

$p r^{3} = 3 \cdot 9$

Step 3.2.1.2

Multiply $3$ by $9$ .

$p r^{3} = 27$

Step 4

Divide each term in $p r^{3} = 27$ by $p$ and simplify.

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Step 4.1

Divide each term in $p r^{3} = 27$ by $p$ .

$\frac{p r^{3}}{p} = \frac{27}{p}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $p$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{p r^{3}}{p} = \frac{27}{p}$

Step 4.2.1.2

Divide $r^{3}$ by $1$ .

$r^{3} = \frac{27}{p}$

Step 5

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

$r = \sqrt[3]{\frac{27}{p}}$

Step 6

Simplify $\sqrt[3]{\frac{27}{p}}$ .

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Step 6.1

Rewrite $\sqrt[3]{\frac{27}{p}}$ as $\frac{\sqrt[3]{27}}{\sqrt[3]{p}}$ .

$r = \frac{\sqrt[3]{27}}{\sqrt[3]{p}}$

Step 6.2

Simplify the numerator.

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Step 6.2.1

Rewrite $27$ as $3^{3}$ .

$r = \frac{\sqrt[3]{3^{3}}}{\sqrt[3]{p}}$

Step 6.2.2

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

$r = \frac{3}{\sqrt[3]{p}}$

Step 6.3

Multiply $\frac{3}{\sqrt[3]{p}}$ by $\frac{{\sqrt[3]{p}}^{2}}{{\sqrt[3]{p}}^{2}}$ .

$r = \frac{3}{\sqrt[3]{p}} \cdot \frac{{\sqrt[3]{p}}^{2}}{{\sqrt[3]{p}}^{2}}$

Step 6.4

Combine and simplify the denominator.

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Step 6.4.1

Multiply $\frac{3}{\sqrt[3]{p}}$ by $\frac{{\sqrt[3]{p}}^{2}}{{\sqrt[3]{p}}^{2}}$ .

$r = \frac{3 {\sqrt[3]{p}}^{2}}{\sqrt[3]{p} {\sqrt[3]{p}}^{2}}$

Step 6.4.2

Raise $\sqrt[3]{p}$ to the power of $1$ .

$r = \frac{3 {\sqrt[3]{p}}^{2}}{{\sqrt[3]{p}}^{1} {\sqrt[3]{p}}^{2}}$

Step 6.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$r = \frac{3 {\sqrt[3]{p}}^{2}}{{\sqrt[3]{p}}^{1 + 2}}$

Step 6.4.4

Add $1$ and $2$ .

$r = \frac{3 {\sqrt[3]{p}}^{2}}{{\sqrt[3]{p}}^{3}}$

Step 6.4.5

Rewrite ${\sqrt[3]{p}}^{3}$ as $p$ .

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Step 6.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{p}$ as $p^{\frac{1}{3}}$ .

$r = \frac{3 {\sqrt[3]{p}}^{2}}{{(p^{\frac{1}{3}})}^{3}}$

Step 6.4.5.2

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

$r = \frac{3 {\sqrt[3]{p}}^{2}}{p^{\frac{1}{3} \cdot 3}}$

Step 6.4.5.3

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

$r = \frac{3 {\sqrt[3]{p}}^{2}}{p^{\frac{3}{3}}}$

Step 6.4.5.4

Cancel the common factor of $3$ .

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Step 6.4.5.4.1

Cancel the common factor.

$r = \frac{3 {\sqrt[3]{p}}^{2}}{p^{\frac{3}{3}}}$

Step 6.4.5.4.2

Rewrite the expression.

$r = \frac{3 {\sqrt[3]{p}}^{2}}{p^{1}}$

Step 6.4.5.5

Simplify.

$r = \frac{3 {\sqrt[3]{p}}^{2}}{p}$

Step 6.5

Rewrite ${\sqrt[3]{p}}^{2}$ as $\sqrt[3]{p^{2}}$ .

$r = \frac{3 \sqrt[3]{p^{2}}}{p}$