Algebra Examples

$r = \sqrt{\frac{v}{π h}}$

Step 1

Rewrite the equation as $\sqrt{\frac{v}{π h}} = r$ .

$\sqrt{\frac{v}{π h}} = r$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{\frac{v}{π h}}}^{2} = r^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(\frac{v}{π h})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(\frac{v}{π h})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

${(\frac{v}{π h})}^{1} = r^{2}$

Step 3.2.1.2

Simplify.

$\frac{v}{π h} = r^{2}$

Step 4

Solve for $v$ .

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Step 4.1

Multiply both sides by $π h$ .

$\frac{v}{π h} (π h) = r^{2} (π h)$

Step 4.2

Simplify.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Simplify $\frac{v}{π h} (π h)$ .

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Step 4.2.1.1.1

Rewrite using the commutative property of multiplication.

$π \frac{v}{π h} h = r^{2} (π h)$

Step 4.2.1.1.2

Cancel the common factor of $π$ .

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Step 4.2.1.1.2.1

Factor $π$ out of $π h$ .

$π \frac{v}{π (h)} h = r^{2} (π h)$

Step 4.2.1.1.2.2

Cancel the common factor.

$π \frac{v}{π h} h = r^{2} (π h)$

Step 4.2.1.1.2.3

Rewrite the expression.

$\frac{v}{h} h = r^{2} (π h)$

Step 4.2.1.1.3

Cancel the common factor of $h$ .

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Step 4.2.1.1.3.1

Cancel the common factor.

$\frac{v}{h} h = r^{2} (π h)$

Step 4.2.1.1.3.2

Rewrite the expression.

$v = r^{2} (π h)$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Rewrite using the commutative property of multiplication.

$v = π r^{2} h$