Solve for r v=1/3*(pr^2h)

$v = \frac{1}{3} \cdot (p r^{2} h)$

Step 1

Rewrite the equation as $\frac{1}{3} \cdot (p r^{2} h) = v$ .

$\frac{1}{3} \cdot (p r^{2} h) = v$

Step 2

Multiply both sides of the equation by $3$ .

$3 (\frac{1}{3} \cdot (p r^{2} h)) = 3 v$

Step 3

Simplify the left side.

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

Multiply $\frac{1}{3} (p r^{2} h)$ .

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

$3 (\frac{p}{3} (r^{2} h)) = 3 v$

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

$3 (\frac{r^{2} p}{3} h) = 3 v$

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

$3 \frac{r^{2} p h}{3} = 3 v$

Cancel the common factor of $3$ .

Cancel the common factor.

$3 \frac{r^{2} p h}{3} = 3 v$

Rewrite the expression.

$r^{2} p h = 3 v$

Step 4

Divide each term in $r^{2} p h = 3 v$ by $p h$ and simplify.

Divide each term in $r^{2} p h = 3 v$ by $p h$ .

$\frac{r^{2} p h}{p h} = \frac{3 v}{p h}$

Simplify the left side.

Cancel the common factor of $p$ .

Cancel the common factor.

$\frac{r^{2} p h}{p h} = \frac{3 v}{p h}$

Rewrite the expression.

$\frac{r^{2} h}{h} = \frac{3 v}{p h}$

Cancel the common factor of $h$ .

Cancel the common factor.

$\frac{r^{2} h}{h} = \frac{3 v}{p h}$

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

$r^{2} = \frac{3 v}{p h}$

Step 5

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

$r = \pm \sqrt{\frac{3 v}{p h}}$

Step 6

Simplify $\pm \sqrt{\frac{3 v}{p h}}$ .

Rewrite $\sqrt{\frac{3 v}{p h}}$ as $\frac{\sqrt{3 v}}{\sqrt{p h}}$ .

$r = \pm \frac{\sqrt{3 v}}{\sqrt{p h}}$

Multiply $\frac{\sqrt{3 v}}{\sqrt{p h}}$ by $\frac{\sqrt{p h}}{\sqrt{p h}}$ .

$r = \pm \frac{\sqrt{3 v}}{\sqrt{p h}} \cdot \frac{\sqrt{p h}}{\sqrt{p h}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{3 v}}{\sqrt{p h}}$ by $\frac{\sqrt{p h}}{\sqrt{p h}}$ .

$r = \pm \frac{\sqrt{3 v} \sqrt{p h}}{\sqrt{p h} \sqrt{p h}}$

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

$r = \pm \frac{\sqrt{3 v} \sqrt{p h}}{{\sqrt{p h}}^{1} \sqrt{p h}}$

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

$r = \pm \frac{\sqrt{3 v} \sqrt{p h}}{{\sqrt{p h}}^{1} {\sqrt{p h}}^{1}}$

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

$r = \pm \frac{\sqrt{3 v} \sqrt{p h}}{{\sqrt{p h}}^{1 + 1}}$

Add $1$ and $1$ .

$r = \pm \frac{\sqrt{3 v} \sqrt{p h}}{{\sqrt{p h}}^{2}}$

Rewrite ${\sqrt{p h}}^{2}$ as $p h$ .

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

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

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

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

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

$r = \pm \frac{\sqrt{3 v} \sqrt{p h}}{{(p h)}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$r = \pm \frac{\sqrt{3 v} \sqrt{p h}}{{(p h)}^{\frac{2}{2}}}$

Rewrite the expression.

$r = \pm \frac{\sqrt{3 v} \sqrt{p h}}{{(p h)}^{1}}$

Simplify.

$r = \pm \frac{\sqrt{3 v} \sqrt{p h}}{p h}$

Combine using the product rule for radicals.

$r = \pm \frac{\sqrt{3 v p h}}{p h}$

Step 7

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.

$r = \frac{\sqrt{3 v p h}}{p h}$

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

$r = - \frac{\sqrt{3 v p h}}{p h}$

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

$r = \frac{\sqrt{3 v p h}}{p h}$

$r = - \frac{\sqrt{3 v p h}}{p h}$

$r = \frac{\sqrt{3 v p h}}{p h}$

$r = - \frac{\sqrt{3 v p h}}{p h}$

Solve for r v=1/3*(pr^2h)

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