Solve for r V=4/3*(pr^3)

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

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.

Simplify the left side.

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

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

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

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

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

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

Combine.

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

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

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

Simplify the right side.

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.

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

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

Simplify the left side.

Cancel the common factor of $p$ .

Cancel the common factor.

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

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

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

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the numerator.

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

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

Apply the product rule to $4 p$ .

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

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

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

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

Factor $8$ out of $16$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Combine exponents.

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

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

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

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

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

Cancel the common factors.

Factor $2$ out of $4 p$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Solve for r V=4/3*(pr^3)

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