Algebra Examples

$V = \frac{4}{3} π r^{3}$

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.

$r = \frac{\sqrt[3]{6 V π^{2}}}{2 π}$