Write as a Function of v r = cube root of (3v)/(4pi)

$r = \sqrt[3]{\frac{3 v}{4 π}}$

Step 1

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

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

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the numerator.

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

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

Apply the product rule to $4 π$ .

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

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

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

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

Factor $8$ out of $16$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Combine exponents.

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

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

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

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

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

Cancel the common factors.

Factor $2$ out of $4 π$ .

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

Cancel the common factor.

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

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.

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

Write as a Function of v r = cube root of (3v)/(4pi)

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