Algebra Examples

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

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

Step 1

Rewrite the equation as

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

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

\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

$3$ .

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

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

Step 3

Simplify the left side.

Tap for more steps...

Step 3.1

Simplify

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

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

Tap for more steps...

Step 3.1.1

Multiply

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

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

Tap for more steps...

Step 3.1.1.1

Combine

p

$p$ and

\frac{1}{3}

$\frac{1}{3}$ .

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

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

Step 3.1.1.2

Combine

r^{2}

$r^{2}$ and

\frac{p}{3}

$\frac{p}{3}$ .

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

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

Step 3.1.1.3

Combine

\frac{r^{2} p}{3}

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

h

$h$ .

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

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

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

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

Step 3.1.2

Cancel the common factor of

3

$3$ .

Tap for more steps...

Step 3.1.2.1

Cancel the common factor.

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

Step 3.1.2.2

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.

Tap for more steps...

Step 4.1

Divide each term in

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

$p h$ .

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

Step 4.2

Simplify the left side.

Tap for more steps...

Step 4.2.1

Cancel the common factor of

$p$ .

Tap for more steps...

Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Rewrite the expression.

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

Step 4.2.2

Cancel the common factor of

$h$ .

Tap for more steps...

Step 4.2.2.1

Cancel the common factor.

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

Step 4.2.2.2

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

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

Tap for more steps...

Step 6.1

Rewrite

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

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

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

Step 6.2

Multiply

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

$\frac{\sqrt{p h}}{\sqrt{p h}}$ .

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

Step 6.3

Combine and simplify the denominator.

Tap for more steps...

Step 6.3.1

Multiply

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

$\frac{\sqrt{p h}}{\sqrt{p h}}$ .

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

Step 6.3.2

Raise

$\sqrt{p h}$ to the power of

$1$ .

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

Step 6.3.3

Raise

$\sqrt{p h}$ to the power of

$1$ .

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

Step 6.3.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.3.5

Add

$1$ and

$1$ .

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

Step 6.3.6

Rewrite

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

$p h$ .

Tap for more steps...

Step 6.3.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{p h}$ as

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

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

Step 6.3.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

Step 6.3.6.3

Combine

$\frac{1}{2}$ and

$2$ .

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

Step 6.3.6.4

Cancel the common factor of

$2$ .

Tap for more steps...

Step 6.3.6.4.1

Cancel the common factor.

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

Step 6.3.6.4.2

Rewrite the expression.

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

Step 6.3.6.5

Simplify.

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

Step 6.4

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.

Tap for more steps...

Step 7.1

First, use the positive value of the

$\pm$ to find the first solution.

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

Step 7.2

Next, use the negative value of the

$\pm$ to find the second solution.

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

Step 7.3

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

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

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

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

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