Basic Math Examples

v = \frac{4}{3} π r^{3}

$v = \frac{4}{3} π r^{3}$

Step 1

Rewrite the equation as

$\frac{4}{3} \cdot (π r^{3}) = v$ .

$\frac{4}{3} \cdot (π r^{3}) = v$

Step 2

Multiply both sides of the equation by

$\frac{1}{\frac{4}{3} π}$ .

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

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

$\frac{1}{\frac{4}{3} π} (\frac{4}{3} \cdot (π r^{3}))$ .

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Step 3.1.1.1

Combine

$\frac{4}{3}$ and

$π$ .

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

Step 3.1.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.1.1.3

Multiply

$\frac{3}{4 π}$ by

$1$ .

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

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

$4 π$ .

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

Step 3.1.1.4.2

Factor

$π$ out of

$\frac{4}{3} \cdot (π r^{3})$ .

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

Step 3.1.1.4.3

Cancel the common factor.

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

Step 3.1.1.4.4

Rewrite the expression.

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

Step 3.1.1.5

Combine

$\frac{4}{3}$ and

$r^{3}$ .

$\frac{3}{4} \cdot \frac{4 r^{3}}{3} = \frac{1}{\frac{4}{3} π} v$

Step 3.1.1.6

Multiply

$\frac{3}{4}$ by

$\frac{4 r^{3}}{3}$ .

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

Step 3.1.1.7

Multiply.

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Step 3.1.1.7.1

Multiply

$4$ by

$3$ .

$\frac{12 r^{3}}{4 \cdot 3} = \frac{1}{\frac{4}{3} π} v$

Step 3.1.1.7.2

Multiply

$4$ by

$3$ .

$\frac{12 r^{3}}{12} = \frac{1}{\frac{4}{3} π} v$

Step 3.1.1.8

Cancel the common factor of

$12$ .

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Step 3.1.1.8.1

Cancel the common factor.

$\frac{12 r^{3}}{12} = \frac{1}{\frac{4}{3} π} v$

Step 3.1.1.8.2

Divide

$r^{3}$ by

$1$ .

$r^{3} = \frac{1}{\frac{4}{3} π} v$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify

$\frac{1}{\frac{4}{3} π} v$ .

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Step 3.2.1.1

Combine

$\frac{4}{3}$ and

$π$ .

$r^{3} = \frac{1}{\frac{4 π}{3}} v$

Step 3.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$r^{3} = 1 \frac{3}{4 π} v$

Step 3.2.1.3

Multiply

$\frac{3}{4 π}$ by

$1$ .

$r^{3} = \frac{3}{4 π} v$

Step 3.2.1.4

Combine

$\frac{3}{4 π}$ and

$v$ .

$r^{3} = \frac{3 v}{4 π}$

Step 4

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

$r = \sqrt[3]{\frac{3 v}{4 π}}$

Step 5

Simplify

$\sqrt[3]{\frac{3 v}{4 π}}$ .

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Step 5.1

Rewrite

$\sqrt[3]{\frac{3 v}{4 π}}$ as

$\frac{\sqrt[3]{3 v}}{\sqrt[3]{4 π}}$ .

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

Step 5.2

Multiply

$\frac{\sqrt[3]{3 v}}{\sqrt[3]{4 π}}$ by

$\frac{{\sqrt[3]{4 π}}^{2}}{{\sqrt[3]{4 π}}^{2}}$ .

$r = \frac{\sqrt[3]{3 v}}{\sqrt[3]{4 π}} \cdot \frac{{\sqrt[3]{4 π}}^{2}}{{\sqrt[3]{4 π}}^{2}}$

Step 5.3

Combine and simplify the denominator.

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Step 5.3.1

Multiply

$\frac{\sqrt[3]{3 v}}{\sqrt[3]{4 π}}$ by

$\frac{{\sqrt[3]{4 π}}^{2}}{{\sqrt[3]{4 π}}^{2}}$ .

$r = \frac{\sqrt[3]{3 v} {\sqrt[3]{4 π}}^{2}}{\sqrt[3]{4 π} {\sqrt[3]{4 π}}^{2}}$

Step 5.3.2

Raise

$\sqrt[3]{4 π}$ to the power of

$1$ .

$r = \frac{\sqrt[3]{3 v} {\sqrt[3]{4 π}}^{2}}{{\sqrt[3]{4 π}}^{1} {\sqrt[3]{4 π}}^{2}}$

Step 5.3.3

Use the power rule

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

$r = \frac{\sqrt[3]{3 v} {\sqrt[3]{4 π}}^{2}}{{\sqrt[3]{4 π}}^{1 + 2}}$

Step 5.3.4

Add

$1$ and

$2$ .

$r = \frac{\sqrt[3]{3 v} {\sqrt[3]{4 π}}^{2}}{{\sqrt[3]{4 π}}^{3}}$

Step 5.3.5

Rewrite

${\sqrt[3]{4 π}}^{3}$ as

$4 π$ .

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Step 5.3.5.1

Use

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

$\sqrt[3]{4 π}$ as

${(4 π)}^{\frac{1}{3}}$ .

$r = \frac{\sqrt[3]{3 v} {\sqrt[3]{4 π}}^{2}}{{({(4 π)}^{\frac{1}{3}})}^{3}}$

Step 5.3.5.2

Apply the power rule and multiply exponents,

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

$r = \frac{\sqrt[3]{3 v} {\sqrt[3]{4 π}}^{2}}{{(4 π)}^{\frac{1}{3} \cdot 3}}$

Step 5.3.5.3

Combine

$\frac{1}{3}$ and

$3$ .

$r = \frac{\sqrt[3]{3 v} {\sqrt[3]{4 π}}^{2}}{{(4 π)}^{\frac{3}{3}}}$

Step 5.3.5.4

Cancel the common factor of

$3$ .

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Step 5.3.5.4.1

Cancel the common factor.

$r = \frac{\sqrt[3]{3 v} {\sqrt[3]{4 π}}^{2}}{{(4 π)}^{\frac{3}{3}}}$

Step 5.3.5.4.2

Rewrite the expression.

$r = \frac{\sqrt[3]{3 v} {\sqrt[3]{4 π}}^{2}}{{(4 π)}^{1}}$

Step 5.3.5.5

Simplify.

$r = \frac{\sqrt[3]{3 v} {\sqrt[3]{4 π}}^{2}}{4 π}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Rewrite

${\sqrt[3]{4 π}}^{2}$ as

$\sqrt[3]{{(4 π)}^{2}}$ .

$r = \frac{\sqrt[3]{3 v} \sqrt[3]{{(4 π)}^{2}}}{4 π}$

Step 5.4.2

Apply the product rule to

$4 π$ .

$r = \frac{\sqrt[3]{3 v} \sqrt[3]{4^{2} π^{2}}}{4 π}$

Step 5.4.3

Raise

$4$ to the power of

$2$ .

$r = \frac{\sqrt[3]{3 v} \sqrt[3]{16 π^{2}}}{4 π}$

Step 5.4.4

Rewrite

$16 π^{2}$ as

$2^{3} \cdot (2 π^{2})$ .

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Step 5.4.4.1

Factor

$8$ out of

$16$ .

$r = \frac{\sqrt[3]{3 v} \sqrt[3]{8 (2) π^{2}}}{4 π}$

Step 5.4.4.2

Rewrite

$8$ as

$2^{3}$ .

$r = \frac{\sqrt[3]{3 v} \sqrt[3]{2^{3} \cdot 2 π^{2}}}{4 π}$

Step 5.4.4.3

Add parentheses.

$r = \frac{\sqrt[3]{3 v} \sqrt[3]{2^{3} \cdot (2 π^{2})}}{4 π}$

Step 5.4.5

Pull terms out from under the radical.

$r = \frac{\sqrt[3]{3 v} \cdot 2 \sqrt[3]{2 π^{2}}}{4 π}$

Step 5.4.6

Combine exponents.

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Step 5.4.6.1

Combine using the product rule for radicals.

$r = \frac{2 \sqrt[3]{3 v (2 π^{2})}}{4 π}$

Step 5.4.6.2

Multiply

$2$ by

$3$ .

$r = \frac{2 \sqrt[3]{6 v π^{2}}}{4 π}$

Step 5.5

Cancel the common factor of

$2$ and

$4$ .

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Step 5.5.1

Factor

$2$ out of

$2 \sqrt[3]{6 v π^{2}}$ .

$r = \frac{2 (\sqrt[3]{6 v π^{2}})}{4 π}$

Step 5.5.2

Cancel the common factors.

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Step 5.5.2.1

Factor

$2$ out of

$4 π$ .

$r = \frac{2 (\sqrt[3]{6 v π^{2}})}{2 (2 π)}$

Step 5.5.2.2

Cancel the common factor.

$r = \frac{2 \sqrt[3]{6 v π^{2}}}{2 (2 π)}$

Step 5.5.2.3

Rewrite the expression.

$r = \frac{\sqrt[3]{6 v π^{2}}}{2 π}$