Solve for r v=4/3pir^2

$v = \frac{4}{3} π r^{2}$

Step 1

Rewrite the equation as $\frac{4}{3} \cdot (π r^{2}) = v$ .

$\frac{4}{3} \cdot (π r^{2}) = v$

Step 2

Multiply both sides of the equation by $\frac{1}{\frac{4}{3} π}$ .

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

Step 3

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{1}{\frac{4}{3} π} (\frac{4}{3} \cdot (π r^{2}))$ .

Combine $\frac{4}{3}$ and $π$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $\frac{3}{4 π}$ by $1$ .

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

Cancel the common factor of $π$ .

Factor $π$ out of $4 π$ .

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

Factor $π$ out of $\frac{4}{3} \cdot (π r^{2})$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Combine $\frac{4}{3}$ and $r^{2}$ .

$\frac{3}{4} \cdot \frac{4 r^{2}}{3} = \frac{1}{\frac{4}{3} π} v$

Multiply $\frac{3}{4}$ by $\frac{4 r^{2}}{3}$ .

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

Multiply.

Multiply $4$ by $3$ .

$\frac{12 r^{2}}{4 \cdot 3} = \frac{1}{\frac{4}{3} π} v$

Multiply $4$ by $3$ .

$\frac{12 r^{2}}{12} = \frac{1}{\frac{4}{3} π} v$

Cancel the common factor of $12$ .

Cancel the common factor.

$\frac{12 r^{2}}{12} = \frac{1}{\frac{4}{3} π} v$

Divide $r^{2}$ by $1$ .

$r^{2} = \frac{1}{\frac{4}{3} π} v$

Simplify the right side.

Simplify $\frac{1}{\frac{4}{3} π} v$ .

Combine $\frac{4}{3}$ and $π$ .

$r^{2} = \frac{1}{\frac{4 π}{3}} v$

Multiply the numerator by the reciprocal of the denominator.

$r^{2} = 1 \frac{3}{4 π} v$

Multiply $\frac{3}{4 π}$ by $1$ .

$r^{2} = \frac{3}{4 π} v$

Combine $\frac{3}{4 π}$ and $v$ .

$r^{2} = \frac{3 v}{4 π}$

Step 4

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

$r = \pm \sqrt{\frac{3 v}{4 π}}$

Step 5

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

Rewrite $\frac{3 v}{4 π}$ as ${(\frac{1}{2})}^{2} \frac{3 v}{π}$ .

Factor the perfect power $1^{2}$ out of $3 v$ .

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

Factor the perfect power $2^{2}$ out of $4 π$ .

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

Rearrange the fraction $\frac{1^{2} (3 v)}{2^{2} π}$ .

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

Pull terms out from under the radical.

$r = \pm \frac{1}{2} \sqrt{\frac{3 v}{π}}$

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

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{3 v}}{\sqrt{π}}$

Multiply $\frac{\sqrt{3 v}}{\sqrt{π}}$ by $\frac{\sqrt{π}}{\sqrt{π}}$ .

$r = \pm \frac{1}{2} (\frac{\sqrt{3 v}}{\sqrt{π}} \cdot \frac{\sqrt{π}}{\sqrt{π}})$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{3 v}}{\sqrt{π}}$ by $\frac{\sqrt{π}}{\sqrt{π}}$ .

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{3 v} \sqrt{π}}{\sqrt{π} \sqrt{π}}$

Raise $\sqrt{π}$ to the power of $1$ .

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{3 v} \sqrt{π}}{{\sqrt{π}}^{1} \sqrt{π}}$

Raise $\sqrt{π}$ to the power of $1$ .

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{3 v} \sqrt{π}}{{\sqrt{π}}^{1} {\sqrt{π}}^{1}}$

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

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{3 v} \sqrt{π}}{{\sqrt{π}}^{1 + 1}}$

Add $1$ and $1$ .

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{3 v} \sqrt{π}}{{\sqrt{π}}^{2}}$

Rewrite ${\sqrt{π}}^{2}$ as $π$ .

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

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

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

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{3 v} \sqrt{π}}{π^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{3 v} \sqrt{π}}{π^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{3 v} \sqrt{π}}{π^{\frac{2}{2}}}$

Rewrite the expression.

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{3 v} \sqrt{π}}{π^{1}}$

Simplify.

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{3 v} \sqrt{π}}{π}$

Combine using the product rule for radicals.

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{3 v π}}{π}$

Multiply $\frac{1}{2}$ by $\frac{\sqrt{3 v π}}{π}$ .

$r = \pm \frac{\sqrt{3 v π}}{2 π}$

Step 6

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$r = \frac{\sqrt{3 v π}}{2 π}$

Next, use the negative value of the $\pm$ to find the second solution.

$r = - \frac{\sqrt{3 v π}}{2 π}$

The complete solution is the result of both the positive and negative portions of the solution.

$r = \frac{\sqrt{3 v π}}{2 π}$

$r = - \frac{\sqrt{3 v π}}{2 π}$

$r = \frac{\sqrt{3 v π}}{2 π}$

$r = - \frac{\sqrt{3 v π}}{2 π}$

Solve for r v=4/3pir^2

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