Algebra Examples

$v (r) = \frac{4}{3} π r^{3}$

Step 1

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

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Step 1.1

Multiply $\frac{4}{3} (π r^{3})$ .

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Step 1.1.1

Combine $π$ and $\frac{4}{3}$ .

$v r = \frac{π \cdot 4}{3} r^{3}$

Step 1.1.2

Combine $\frac{π \cdot 4}{3}$ and $r^{3}$ .

$v r = \frac{π \cdot 4 r^{3}}{3}$

Step 1.2

Move $4$ to the left of $π$ .

$v r = \frac{4 π r^{3}}{3}$

Step 2

Subtract $\frac{4 π r^{3}}{3}$ from both sides of the equation.

$v r - \frac{4 π r^{3}}{3} = 0$

Step 3

Factor $r$ out of $v r - \frac{4 π r^{3}}{3}$ .

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Step 3.1

Factor $r$ out of $v r$ .

$r v - \frac{4 π r^{3}}{3} = 0$

Step 3.2

Factor $r$ out of $- \frac{4 π r^{3}}{3}$ .

$r v + r (- \frac{4 π r^{2}}{3}) = 0$

Step 3.3

Factor $r$ out of $r v + r (- \frac{4 π r^{2}}{3})$ .

$r (v - \frac{4 π r^{2}}{3}) = 0$

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$r = 0$

$v - \frac{4 π r^{2}}{3} = 0$

Step 5

Set $r$ equal to $0$ .

$r = 0$

Step 6

Set $v - \frac{4 π r^{2}}{3}$ equal to $0$ and solve for $r$ .

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Step 6.1

Set $v - \frac{4 π r^{2}}{3}$ equal to $0$ .

$v - \frac{4 π r^{2}}{3} = 0$

Step 6.2

Solve $v - \frac{4 π r^{2}}{3} = 0$ for $r$ .

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Step 6.2.1

Subtract $v$ from both sides of the equation.

$- \frac{4 π r^{2}}{3} = - v$

Step 6.2.2

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

$- \frac{3}{4 π} (- \frac{4 π r^{2}}{3}) = - \frac{3}{4 π} (- v)$

Step 6.2.3

Simplify both sides of the equation.

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Step 6.2.3.1

Simplify the left side.

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Step 6.2.3.1.1

Simplify $- \frac{3}{4 π} (- \frac{4 π r^{2}}{3})$ .

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Step 6.2.3.1.1.1

Cancel the common factor of $3$ .

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Step 6.2.3.1.1.1.1

Move the leading negative in $- \frac{3}{4 π}$ into the numerator.

$\frac{- 3}{4 π} (- \frac{4 π r^{2}}{3}) = - \frac{3}{4 π} (- v)$

Step 6.2.3.1.1.1.2

Move the leading negative in $- \frac{4 π r^{2}}{3}$ into the numerator.

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

Step 6.2.3.1.1.1.3

Factor $3$ out of $- 3$ .

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

Step 6.2.3.1.1.1.4

Cancel the common factor.

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

Step 6.2.3.1.1.1.5

Rewrite the expression.

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

Step 6.2.3.1.1.2

Cancel the common factor of $4 π$ .

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Step 6.2.3.1.1.2.1

Factor $4 π$ out of $- 4 π r^{2}$ .

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

Step 6.2.3.1.1.2.2

Cancel the common factor.

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

Step 6.2.3.1.1.2.3

Rewrite the expression.

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

Step 6.2.3.1.1.3

Multiply.

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Step 6.2.3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 r^{2} = - \frac{3}{4 π} (- v)$

Step 6.2.3.1.1.3.2

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

$r^{2} = - \frac{3}{4 π} (- v)$

Step 6.2.3.2

Simplify the right side.

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Step 6.2.3.2.1

Multiply $- \frac{3}{4 π} (- v)$ .

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Step 6.2.3.2.1.1

Multiply $- 1$ by $- 1$ .

$r^{2} = 1 \frac{3}{4 π} v$

Step 6.2.3.2.1.2

Multiply $\frac{3}{4 π}$ by $1$ .

$r^{2} = \frac{3}{4 π} v$

Step 6.2.3.2.1.3

Combine $\frac{3}{4 π}$ and $v$ .

$r^{2} = \frac{3 v}{4 π}$

Step 6.2.4

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

$r = \pm \sqrt{\frac{3 v}{4 π}}$

Step 6.2.5

Simplify $\pm \sqrt{\frac{3 v}{4 π}}$ .

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Step 6.2.5.1

Rewrite $\frac{3 v}{4 π}$ as ${(\frac{1}{2})}^{2} \frac{3 v}{π}$ .

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Step 6.2.5.1.1

Factor the perfect power $1^{2}$ out of $3 v$ .

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

Step 6.2.5.1.2

Factor the perfect power $2^{2}$ out of $4 π$ .

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

Step 6.2.5.1.3

Rearrange the fraction $\frac{1^{2} (3 v)}{2^{2} π}$ .

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

Step 6.2.5.2

Pull terms out from under the radical.

$r = \pm \frac{1}{2} \sqrt{\frac{3 v}{π}}$

Step 6.2.5.3

Rewrite $\sqrt{\frac{3 v}{π}}$ as $\frac{\sqrt{3 v}}{\sqrt{π}}$ .

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{3 v}}{\sqrt{π}}$

Step 6.2.5.4

Multiply $\frac{\sqrt{3 v}}{\sqrt{π}}$ by $\frac{\sqrt{π}}{\sqrt{π}}$ .

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

Step 6.2.5.5

Combine and simplify the denominator.

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Step 6.2.5.5.1

Multiply $\frac{\sqrt{3 v}}{\sqrt{π}}$ by $\frac{\sqrt{π}}{\sqrt{π}}$ .

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{3 v} \sqrt{π}}{\sqrt{π} \sqrt{π}}$

Step 6.2.5.5.2

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

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{3 v} \sqrt{π}}{{\sqrt{π}}^{1} \sqrt{π}}$

Step 6.2.5.5.3

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

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{3 v} \sqrt{π}}{{\sqrt{π}}^{1} {\sqrt{π}}^{1}}$

Step 6.2.5.5.4

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

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{3 v} \sqrt{π}}{{\sqrt{π}}^{1 + 1}}$

Step 6.2.5.5.5

Add $1$ and $1$ .

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{3 v} \sqrt{π}}{{\sqrt{π}}^{2}}$

Step 6.2.5.5.6

Rewrite ${\sqrt{π}}^{2}$ as $π$ .

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Step 6.2.5.5.6.1

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

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

Step 6.2.5.5.6.2

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

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{3 v} \sqrt{π}}{π^{\frac{1}{2} \cdot 2}}$

Step 6.2.5.5.6.3

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

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{3 v} \sqrt{π}}{π^{\frac{2}{2}}}$

Step 6.2.5.5.6.4

Cancel the common factor of $2$ .

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Step 6.2.5.5.6.4.1

Cancel the common factor.

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{3 v} \sqrt{π}}{π^{\frac{2}{2}}}$

Step 6.2.5.5.6.4.2

Rewrite the expression.

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{3 v} \sqrt{π}}{π^{1}}$

Step 6.2.5.5.6.5

Simplify.

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{3 v} \sqrt{π}}{π}$

Step 6.2.5.6

Combine using the product rule for radicals.

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{3 v π}}{π}$

Step 6.2.5.7

Multiply $\frac{1}{2}$ by $\frac{\sqrt{3 v π}}{π}$ .

$r = \pm \frac{\sqrt{3 v π}}{2 π}$

Step 6.2.6

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

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Step 6.2.6.1

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