Basic Math Examples

$v = \frac{4}{3} \cdot (p r^{3})$

Step 1

Rewrite the equation as $\frac{4}{3} \cdot (p r^{3}) = v$ .

$\frac{4}{3} \cdot (p r^{3}) = v$

Step 2

Multiply both sides of the equation by $\frac{3}{4}$ .

$\frac{3}{4} (\frac{4}{3} \cdot (p r^{3})) = \frac{3}{4} v$

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 $\frac{3}{4} (\frac{4}{3} \cdot (p r^{3}))$ .

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Step 3.1.1.1

Multiply $\frac{4}{3} (p r^{3})$ .

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Step 3.1.1.1.1

Combine $p$ and $\frac{4}{3}$ .

$\frac{3}{4} (\frac{p \cdot 4}{3} r^{3}) = \frac{3}{4} v$

Step 3.1.1.1.2

Combine $\frac{p \cdot 4}{3}$ and $r^{3}$ .

$\frac{3}{4} \cdot \frac{p \cdot 4 r^{3}}{3} = \frac{3}{4} v$

Step 3.1.1.2

Combine.

$\frac{3 (p \cdot 4 r^{3})}{4 \cdot 3} = \frac{3}{4} v$

Step 3.1.1.3

Cancel the common factor of $3$ .

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Step 3.1.1.3.1

Cancel the common factor.

$\frac{3 (p \cdot 4 r^{3})}{4 \cdot 3} = \frac{3}{4} v$

Step 3.1.1.3.2

Rewrite the expression.

$\frac{p \cdot 4 r^{3}}{4} = \frac{3}{4} v$

Step 3.1.1.4

Cancel the common factor of $4$ .

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Step 3.1.1.4.1

Cancel the common factor.

$\frac{p \cdot 4 r^{3}}{4} = \frac{3}{4} v$

Step 3.1.1.4.2

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

$p r^{3} = \frac{3}{4} v$

Step 3.2

Simplify the right side.

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Step 3.2.1

Combine $\frac{3}{4}$ and $v$ .

$p r^{3} = \frac{3 v}{4}$

Step 4

Divide each term in $p r^{3} = \frac{3 v}{4}$ by $p$ and simplify.

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Step 4.1

Divide each term in $p r^{3} = \frac{3 v}{4}$ by $p$ .

$\frac{p r^{3}}{p} = \frac{\frac{3 v}{4}}{p}$

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.

$\frac{p r^{3}}{p} = \frac{\frac{3 v}{4}}{p}$

Step 4.2.1.2

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

$r^{3} = \frac{\frac{3 v}{4}}{p}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Multiply the numerator by the reciprocal of the denominator.

$r^{3} = \frac{3 v}{4} \cdot \frac{1}{p}$

Step 4.3.2

Multiply $\frac{3 v}{4}$ by $\frac{1}{p}$ .

$r^{3} = \frac{3 v}{4 p}$

Step 5

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

$r = \sqrt[3]{\frac{3 v}{4 p}}$

Step 6

Simplify $\sqrt[3]{\frac{3 v}{4 p}}$ .

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Step 6.1

Rewrite $\sqrt[3]{\frac{3 v}{4 p}}$ as $\frac{\sqrt[3]{3 v}}{\sqrt[3]{4 p}}$ .

$r = \frac{\sqrt[3]{3 v}}{\sqrt[3]{4 p}}$

Step 6.2

Multiply $\frac{\sqrt[3]{3 v}}{\sqrt[3]{4 p}}$ by $\frac{{\sqrt[3]{4 p}}^{2}}{{\sqrt[3]{4 p}}^{2}}$ .

$r = \frac{\sqrt[3]{3 v}}{\sqrt[3]{4 p}} \cdot \frac{{\sqrt[3]{4 p}}^{2}}{{\sqrt[3]{4 p}}^{2}}$

Step 6.3

Combine and simplify the denominator.

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Step 6.3.1

Multiply $\frac{\sqrt[3]{3 v}}{\sqrt[3]{4 p}}$ by $\frac{{\sqrt[3]{4 p}}^{2}}{{\sqrt[3]{4 p}}^{2}}$ .

$r = \frac{\sqrt[3]{3 v} {\sqrt[3]{4 p}}^{2}}{\sqrt[3]{4 p} {\sqrt[3]{4 p}}^{2}}$

Step 6.3.2

Raise $\sqrt[3]{4 p}$ to the power of $1$ .

$r = \frac{\sqrt[3]{3 v} {\sqrt[3]{4 p}}^{2}}{{\sqrt[3]{4 p}}^{1} {\sqrt[3]{4 p}}^{2}}$

Step 6.3.3

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

$r = \frac{\sqrt[3]{3 v} {\sqrt[3]{4 p}}^{2}}{{\sqrt[3]{4 p}}^{1 + 2}}$

Step 6.3.4

Add $1$ and $2$ .

$r = \frac{\sqrt[3]{3 v} {\sqrt[3]{4 p}}^{2}}{{\sqrt[3]{4 p}}^{3}}$

Step 6.3.5

Rewrite ${\sqrt[3]{4 p}}^{3}$ as $4 p$ .

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Step 6.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 p}$ as ${(4 p)}^{\frac{1}{3}}$ .

$r = \frac{\sqrt[3]{3 v} {\sqrt[3]{4 p}}^{2}}{{({(4 p)}^{\frac{1}{3}})}^{3}}$

Step 6.3.5.2

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

$r = \frac{\sqrt[3]{3 v} {\sqrt[3]{4 p}}^{2}}{{(4 p)}^{\frac{1}{3} \cdot 3}}$

Step 6.3.5.3

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

$r = \frac{\sqrt[3]{3 v} {\sqrt[3]{4 p}}^{2}}{{(4 p)}^{\frac{3}{3}}}$

Step 6.3.5.4

Cancel the common factor of $3$ .

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Step 6.3.5.4.1

Cancel the common factor.

$r = \frac{\sqrt[3]{3 v} {\sqrt[3]{4 p}}^{2}}{{(4 p)}^{\frac{3}{3}}}$

Step 6.3.5.4.2

Rewrite the expression.

$r = \frac{\sqrt[3]{3 v} {\sqrt[3]{4 p}}^{2}}{{(4 p)}^{1}}$

Step 6.3.5.5

Simplify.

$r = \frac{\sqrt[3]{3 v} {\sqrt[3]{4 p}}^{2}}{4 p}$

Step 6.4

Simplify the numerator.

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Step 6.4.1

Rewrite ${\sqrt[3]{4 p}}^{2}$ as $\sqrt[3]{{(4 p)}^{2}}$ .

$r = \frac{\sqrt[3]{3 v} \sqrt[3]{{(4 p)}^{2}}}{4 p}$

Step 6.4.2

Apply the product rule to $4 p$ .

$r = \frac{\sqrt[3]{3 v} \sqrt[3]{4^{2} p^{2}}}{4 p}$

Step 6.4.3

Raise $4$ to the power of $2$ .

$r = \frac{\sqrt[3]{3 v} \sqrt[3]{16 p^{2}}}{4 p}$

Step 6.4.4

Rewrite $16 p^{2}$ as $2^{3} \cdot (2 p^{2})$ .

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Step 6.4.4.1

Factor $8$ out of $16$ .

$r = \frac{\sqrt[3]{3 v} \sqrt[3]{8 (2) p^{2}}}{4 p}$

Step 6.4.4.2

Rewrite $8$ as $2^{3}$ .

$r = \frac{\sqrt[3]{3 v} \sqrt[3]{2^{3} \cdot 2 p^{2}}}{4 p}$

Step 6.4.4.3

Add parentheses.

$r = \frac{\sqrt[3]{3 v} \sqrt[3]{2^{3} \cdot (2 p^{2})}}{4 p}$

Step 6.4.5

Pull terms out from under the radical.

$r = \frac{\sqrt[3]{3 v} \cdot 2 \sqrt[3]{2 p^{2}}}{4 p}$

Step 6.4.6

Combine exponents.

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Step 6.4.6.1

Combine using the product rule for radicals.

$r = \frac{2 \sqrt[3]{3 v (2 p^{2})}}{4 p}$

Step 6.4.6.2

Multiply $2$ by $3$ .

$r = \frac{2 \sqrt[3]{6 v p^{2}}}{4 p}$

Step 6.5

Cancel the common factor of $2$ and $4$ .

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Step 6.5.1

Factor $2$ out of $2 \sqrt[3]{6 v p^{2}}$ .

$r = \frac{2 (\sqrt[3]{6 v p^{2}})}{4 p}$

Step 6.5.2

Cancel the common factors.

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Step 6.5.2.1

Factor $2$ out of $4 p$ .

$r = \frac{2 (\sqrt[3]{6 v p^{2}})}{2 (2 p)}$

Step 6.5.2.2

Cancel the common factor.

$r = \frac{2 \sqrt[3]{6 v p^{2}}}{2 (2 p)}$

Step 6.5.2.3

Rewrite the expression.

$r = \frac{\sqrt[3]{6 v p^{2}}}{2 p}$