Solve for r x=2/3pir^3 for r

$x = \frac{2}{3} π r^{3}$ for $r$

Step 1

Rewrite the equation as $\frac{2}{3} \cdot (π r^{3}) = x$ .

$\frac{2}{3} \cdot (π r^{3}) = x$

Step 2

Multiply both sides of the equation by $\frac{1}{\frac{2}{3} π}$ .

$\frac{1}{\frac{2}{3} π} (\frac{2}{3} \cdot (π r^{3})) = \frac{1}{\frac{2}{3} π} x$

Step 3

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{1}{\frac{2}{3} π} (\frac{2}{3} \cdot (π r^{3}))$ .

Combine $\frac{2}{3}$ and $π$ .

$\frac{1}{\frac{2 π}{3}} (\frac{2}{3} \cdot (π r^{3})) = \frac{1}{\frac{2}{3} π} x$

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{3}{2 π} (\frac{2}{3} \cdot (π r^{3})) = \frac{1}{\frac{2}{3} π} x$

Multiply $\frac{3}{2 π}$ by $1$ .

$\frac{3}{2 π} (\frac{2}{3} \cdot (π r^{3})) = \frac{1}{\frac{2}{3} π} x$

Cancel the common factor of $π$ .

Factor $π$ out of $2 π$ .

$\frac{3}{π \cdot 2} (\frac{2}{3} \cdot (π r^{3})) = \frac{1}{\frac{2}{3} π} x$

Factor $π$ out of $\frac{2}{3} \cdot (π r^{3})$ .

$\frac{3}{π \cdot 2} (π (\frac{2}{3} \cdot (r^{3}))) = \frac{1}{\frac{2}{3} π} x$

Cancel the common factor.

$\frac{3}{π \cdot 2} (π (\frac{2}{3} \cdot (r^{3}))) = \frac{1}{\frac{2}{3} π} x$

Rewrite the expression.

$\frac{3}{2} (\frac{2}{3} \cdot (r^{3})) = \frac{1}{\frac{2}{3} π} x$

Combine $\frac{2}{3}$ and $r^{3}$ .

$\frac{3}{2} \cdot \frac{2 r^{3}}{3} = \frac{1}{\frac{2}{3} π} x$

Multiply $\frac{3}{2}$ by $\frac{2 r^{3}}{3}$ .

$\frac{3 (2 r^{3})}{2 \cdot 3} = \frac{1}{\frac{2}{3} π} x$

Multiply.

Multiply $2$ by $3$ .

$\frac{6 r^{3}}{2 \cdot 3} = \frac{1}{\frac{2}{3} π} x$

Multiply $2$ by $3$ .

$\frac{6 r^{3}}{6} = \frac{1}{\frac{2}{3} π} x$

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 r^{3}}{6} = \frac{1}{\frac{2}{3} π} x$

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

$r^{3} = \frac{1}{\frac{2}{3} π} x$

Simplify the right side.

Simplify $\frac{1}{\frac{2}{3} π} x$ .

Combine $\frac{2}{3}$ and $π$ .

$r^{3} = \frac{1}{\frac{2 π}{3}} x$

Multiply the numerator by the reciprocal of the denominator.

$r^{3} = 1 \frac{3}{2 π} x$

Multiply $\frac{3}{2 π}$ by $1$ .

$r^{3} = \frac{3}{2 π} x$

Combine $\frac{3}{2 π}$ and $x$ .

$r^{3} = \frac{3 x}{2 π}$

Step 4

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

$r = \sqrt[3]{\frac{3 x}{2 π}}$

Step 5

Simplify $\sqrt[3]{\frac{3 x}{2 π}}$ .

Rewrite $\sqrt[3]{\frac{3 x}{2 π}}$ as $\frac{\sqrt[3]{3 x}}{\sqrt[3]{2 π}}$ .

$r = \frac{\sqrt[3]{3 x}}{\sqrt[3]{2 π}}$

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

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

Combine and simplify the denominator.

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

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

Raise $\sqrt[3]{2 π}$ to the power of $1$ .

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

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

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

Add $1$ and $2$ .

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

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2 π}$ as ${(2 π)}^{\frac{1}{3}}$ .

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the numerator.

Rewrite ${\sqrt[3]{2 π}}^{2}$ as ${({(2 π)}^{2})}^{\frac{1}{3}}$ .

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

Apply the product rule to $2 π$ .

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

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

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

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $4$ by $3$ .

$r = \frac{\sqrt[3]{12 x π^{2}}}{2 π}$

Solve for r x=2/3pir^3 for r

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