Solve for V r = cube root of (3V)/(4pi)

$r = \sqrt[3]{\frac{3 V}{4 π}}$

Step 1

Rewrite the equation as $\sqrt[3]{\frac{3 V}{4 π}} = r$ .

$\sqrt[3]{\frac{3 V}{4 π}} = r$

Step 2

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

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

Step 3

Simplify each side of the equation.

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

${({(\frac{3 V}{4 π})}^{\frac{1}{3}})}^{3} = r^{3}$

Simplify the left side.

Simplify ${({(\frac{3 V}{4 π})}^{\frac{1}{3}})}^{3}$ .

Multiply the exponents in ${({(\frac{3 V}{4 π})}^{\frac{1}{3}})}^{3}$ .

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

${(\frac{3 V}{4 π})}^{\frac{1}{3} \cdot 3} = r^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(\frac{3 V}{4 π})}^{\frac{1}{3} \cdot 3} = r^{3}$

Rewrite the expression.

${(\frac{3 V}{4 π})}^{1} = r^{3}$

Simplify.

$\frac{3 V}{4 π} = r^{3}$

Step 4

Solve for $V$ .

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

$\frac{4 π}{3} \cdot \frac{3 V}{4 π} = \frac{4 π}{3} r^{3}$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{4 π}{3} \cdot \frac{3 V}{4 π}$ .

Cancel the common factor of $4 π$ .

Cancel the common factor.

$\frac{4 π}{3} \cdot \frac{3 V}{4 π} = \frac{4 π}{3} r^{3}$

Rewrite the expression.

$\frac{1}{3} (3 V) = \frac{4 π}{3} r^{3}$

Cancel the common factor of $3$ .

Factor $3$ out of $3 V$ .

$\frac{1}{3} (3 (V)) = \frac{4 π}{3} r^{3}$

Cancel the common factor.

$\frac{1}{3} (3 V) = \frac{4 π}{3} r^{3}$

Rewrite the expression.

$V = \frac{4 π}{3} r^{3}$

Simplify the right side.

Combine $\frac{4 π}{3}$ and $r^{3}$ .

$V = \frac{4 π r^{3}}{3}$