Algebra Examples

$r = \sqrt[3]{\frac{3 v}{4 π}}$

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

Add $1$ and $2$ .

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

Step 1.3.5

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

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Step 1.3.5.1

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

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

Step 1.3.5.2

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

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

Step 1.3.5.3

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

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

Step 1.3.5.4

Cancel the common factor of $3$ .

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Step 1.3.5.4.1

Cancel the common factor.

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

Step 1.3.5.4.2

Rewrite the expression.

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

Step 1.3.5.5

Simplify.

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

Step 1.4

Simplify the numerator.

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Step 1.4.1

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

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

Step 1.4.2

Apply the product rule to $4 π$ .

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

Step 1.4.3

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

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

Step 1.4.4

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

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Step 1.4.4.1

Factor $8$ out of $16$ .

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

Step 1.4.4.2

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

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

Step 1.4.4.3

Add parentheses.

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

Step 1.4.5

Pull terms out from under the radical.

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

Step 1.4.6

Combine exponents.

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Step 1.4.6.1

Combine using the product rule for radicals.

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

Step 1.4.6.2

Multiply $2$ by $3$ .

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

Step 1.5

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

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Step 1.5.1

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

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

Step 1.5.2

Cancel the common factors.

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Step 1.5.2.1

Factor $2$ out of $4 π$ .

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

Step 1.5.2.2

Cancel the common factor.

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

Step 1.5.2.3

Rewrite the expression.

$r = \frac{\sqrt[3]{6 v π^{2}}}{2 π}$

Step 2

To rewrite as a function of $v$ , write the equation so that $r$ is by itself on one side of the equal sign and an expression involving only $v$ is on the other side.

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