Algebra Examples

$32 = \sqrt[3]{\frac{3 v}{4 π}}$

Step 1

Rewrite the equation as $\sqrt[3]{\frac{3 v}{4 π}} = 32$ .

$\sqrt[3]{\frac{3 v}{4 π}} = 32$

Step 2

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

${\sqrt[3]{\frac{3 v}{4 π}}}^{3} = 32^{3}$

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 $\sqrt[3]{\frac{3 v}{4 π}}$ as ${(\frac{3 v}{4 π})}^{\frac{1}{3}}$ .

${({(\frac{3 v}{4 π})}^{\frac{1}{3}})}^{3} = 32^{3}$

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

Multiply the exponents in ${({(\frac{3 v}{4 π})}^{\frac{1}{3}})}^{3}$ .

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Step 3.2.1.1.1

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

${(\frac{3 v}{4 π})}^{\frac{1}{3} \cdot 3} = 32^{3}$

Step 3.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(\frac{3 v}{4 π})}^{\frac{1}{3} \cdot 3} = 32^{3}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(\frac{3 v}{4 π})}^{1} = 32^{3}$

Step 3.2.1.2

Simplify.

$\frac{3 v}{4 π} = 32^{3}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Raise $32$ to the power of $3$ .

$\frac{3 v}{4 π} = 32768$

Step 4

Solve for $v$ .

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Step 4.1

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

$\frac{4 π}{3} \cdot \frac{3 v}{4 π} = \frac{4 π}{3} \cdot 32768$

Step 4.2

Simplify both sides of the equation.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

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

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Step 4.2.1.1.1

Cancel the common factor of $4 π$ .

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Step 4.2.1.1.1.1

Cancel the common factor.

$\frac{4 π}{3} \cdot \frac{3 v}{4 π} = \frac{4 π}{3} \cdot 32768$

Step 4.2.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 v) = \frac{4 π}{3} \cdot 32768$

Step 4.2.1.1.2

Cancel the common factor of $3$ .

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Step 4.2.1.1.2.1

Factor $3$ out of $3 v$ .

$\frac{1}{3} (3 (v)) = \frac{4 π}{3} \cdot 32768$

Step 4.2.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 v) = \frac{4 π}{3} \cdot 32768$

Step 4.2.1.1.2.3

Rewrite the expression.

$v = \frac{4 π}{3} \cdot 32768$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Multiply $\frac{4 π}{3} \cdot 32768$ .

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Step 4.2.2.1.1

Combine $\frac{4 π}{3}$ and $32768$ .

$v = \frac{4 π \cdot 32768}{3}$

Step 4.2.2.1.2

Multiply $32768$ by $4$ .

$v = \frac{131072 π}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$v = \frac{131072 π}{3}$

Decimal Form:

$v = 137258.27743044$