Pre-Algebra Examples

$V = \frac{1}{3} \cdot (π r^{2} h)$

Step 1

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

$\frac{1}{3} \cdot (π r^{2} h) = V$

Step 2

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

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

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

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

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Step 3.1.1.1

Combine $\frac{1}{3}$ and $π$ .

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

Step 3.1.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.1.1.3

Multiply $\frac{3}{π}$ by $1$ .

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

Step 3.1.1.4

Cancel the common factor of $π$ .

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Step 3.1.1.4.1

Factor $π$ out of $\frac{1}{3} \cdot (π r^{2} h)$ .

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

Step 3.1.1.4.2

Cancel the common factor.

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

Step 3.1.1.4.3

Rewrite the expression.

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

Step 3.1.1.5

Combine $r^{2}$ and $\frac{1}{3}$ .

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

Step 3.1.1.6

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

$3 \frac{r^{2} h}{3} = \frac{1}{\frac{1}{3} π} V$

Step 3.1.1.7

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

$\frac{3 (r^{2} h)}{3} = \frac{1}{\frac{1}{3} π} V$

Step 3.1.1.8

Cancel the common factor of $3$ .

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Step 3.1.1.8.1

Cancel the common factor.

$\frac{3 (r^{2} h)}{3} = \frac{1}{\frac{1}{3} π} V$

Step 3.1.1.8.2

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

$r^{2} h = \frac{1}{\frac{1}{3} π} V$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\frac{1}{\frac{1}{3} π} V$ .

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Step 3.2.1.1

Combine $\frac{1}{3}$ and $π$ .

$r^{2} h = \frac{1}{\frac{π}{3}} V$

Step 3.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$r^{2} h = 1 \frac{3}{π} V$

Step 3.2.1.3

Multiply $\frac{3}{π}$ by $1$ .

$r^{2} h = \frac{3}{π} V$

Step 3.2.1.4

Combine $\frac{3}{π}$ and $V$ .

$r^{2} h = \frac{3 V}{π}$

Step 4

Divide each term in $r^{2} h = \frac{3 V}{π}$ by $h$ and simplify.

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Step 4.1

Divide each term in $r^{2} h = \frac{3 V}{π}$ by $h$ .

$\frac{r^{2} h}{h} = \frac{\frac{3 V}{π}}{h}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $h$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{r^{2} h}{h} = \frac{\frac{3 V}{π}}{h}$

Step 4.2.1.2

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

$r^{2} = \frac{\frac{3 V}{π}}{h}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3.2

Multiply $\frac{3 V}{π}$ by $\frac{1}{h}$ .

$r^{2} = \frac{3 V}{π h}$

Step 5

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

$r = \pm \sqrt{\frac{3 V}{π h}}$

Step 6

Simplify $\pm \sqrt{\frac{3 V}{π h}}$ .

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Step 6.1

Rewrite $\sqrt{\frac{3 V}{π h}}$ as $\frac{\sqrt{3 V}}{\sqrt{π h}}$ .

$r = \pm \frac{\sqrt{3 V}}{\sqrt{π h}}$

Step 6.2

Multiply $\frac{\sqrt{3 V}}{\sqrt{π h}}$ by $\frac{\sqrt{π h}}{\sqrt{π h}}$ .

$r = \pm \frac{\sqrt{3 V}}{\sqrt{π h}} \cdot \frac{\sqrt{π h}}{\sqrt{π h}}$

Step 6.3

Combine and simplify the denominator.

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Step 6.3.1

Multiply $\frac{\sqrt{3 V}}{\sqrt{π h}}$ by $\frac{\sqrt{π h}}{\sqrt{π h}}$ .

$r = \pm \frac{\sqrt{3 V} \sqrt{π h}}{\sqrt{π h} \sqrt{π h}}$

Step 6.3.2

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

$r = \pm \frac{\sqrt{3 V} \sqrt{π h}}{{\sqrt{π h}}^{1} \sqrt{π h}}$

Step 6.3.3

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

$r = \pm \frac{\sqrt{3 V} \sqrt{π h}}{{\sqrt{π h}}^{1} {\sqrt{π h}}^{1}}$

Step 6.3.4

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

$r = \pm \frac{\sqrt{3 V} \sqrt{π h}}{{\sqrt{π h}}^{1 + 1}}$

Step 6.3.5

Add $1$ and $1$ .

$r = \pm \frac{\sqrt{3 V} \sqrt{π h}}{{\sqrt{π h}}^{2}}$

Step 6.3.6

Rewrite ${\sqrt{π h}}^{2}$ as $π h$ .

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Step 6.3.6.1

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

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

Step 6.3.6.2

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

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

Step 6.3.6.3

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

$r = \pm \frac{\sqrt{3 V} \sqrt{π h}}{{(π h)}^{\frac{2}{2}}}$

Step 6.3.6.4

Cancel the common factor of $2$ .

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Step 6.3.6.4.1

Cancel the common factor.

$r = \pm \frac{\sqrt{3 V} \sqrt{π h}}{{(π h)}^{\frac{2}{2}}}$

Step 6.3.6.4.2

Rewrite the expression.

$r = \pm \frac{\sqrt{3 V} \sqrt{π h}}{{(π h)}^{1}}$

Step 6.3.6.5

Simplify.

$r = \pm \frac{\sqrt{3 V} \sqrt{π h}}{π h}$

Step 6.4

Combine using the product rule for radicals.

$r = \pm \frac{\sqrt{3 V π h}}{π h}$

Step 7

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

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Step 7.1

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

$r = \frac{\sqrt{3 V π h}}{π h}$

Step 7.2

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

$r = - \frac{\sqrt{3 V π h}}{π h}$

Step 7.3

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

$r = \frac{\sqrt{3 V π h}}{π h}$

$r = - \frac{\sqrt{3 V π h}}{π h}$

$r = \frac{\sqrt{3 V π h}}{π h}$

$r = - \frac{\sqrt{3 V π h}}{π h}$