Algebra Examples

$8000 = \frac{4}{3} π r^{3}$

Step 1

Rewrite the equation as $\frac{4}{3} \cdot (π r^{3}) = 8000$ .

$\frac{4}{3} \cdot (π r^{3}) = 8000$

Step 2

Multiply both sides of the equation by $\frac{1}{\frac{4}{3} π}$ .

$\frac{1}{\frac{4}{3} π} (\frac{4}{3} \cdot (π r^{3})) = \frac{1}{\frac{4}{3} π} \cdot 8000$

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 $π$ .

$\frac{1}{\frac{4 π}{3}} (\frac{4}{3} \cdot (π r^{3})) = \frac{1}{\frac{4}{3} π} \cdot 8000$

Step 3.1.1.2

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{3}{4 π} (\frac{4}{3} \cdot (π r^{3})) = \frac{1}{\frac{4}{3} π} \cdot 8000$

Step 3.1.1.3

Multiply $\frac{3}{4 π}$ by $1$ .

$\frac{3}{4 π} (\frac{4}{3} \cdot (π r^{3})) = \frac{1}{\frac{4}{3} π} \cdot 8000$

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 $4 π$ .

$\frac{3}{π \cdot 4} (\frac{4}{3} \cdot (π r^{3})) = \frac{1}{\frac{4}{3} π} \cdot 8000$

Step 3.1.1.4.2

Factor $π$ out of $\frac{4}{3} \cdot (π r^{3})$ .

$\frac{3}{π \cdot 4} (π (\frac{4}{3} \cdot (r^{3}))) = \frac{1}{\frac{4}{3} π} \cdot 8000$

Step 3.1.1.4.3

Cancel the common factor.

$\frac{3}{π \cdot 4} (π (\frac{4}{3} \cdot (r^{3}))) = \frac{1}{\frac{4}{3} π} \cdot 8000$

Step 3.1.1.4.4

Rewrite the expression.

$\frac{3}{4} (\frac{4}{3} \cdot (r^{3})) = \frac{1}{\frac{4}{3} π} \cdot 8000$

Step 3.1.1.5

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

$\frac{3}{4} \cdot \frac{4 r^{3}}{3} = \frac{1}{\frac{4}{3} π} \cdot 8000$

Step 3.1.1.6

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

$\frac{3 (4 r^{3})}{4 \cdot 3} = \frac{1}{\frac{4}{3} π} \cdot 8000$

Step 3.1.1.7

Multiply.

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Step 3.1.1.7.1

Multiply $4$ by $3$ .

$\frac{12 r^{3}}{4 \cdot 3} = \frac{1}{\frac{4}{3} π} \cdot 8000$

Step 3.1.1.7.2

Multiply $4$ by $3$ .

$\frac{12 r^{3}}{12} = \frac{1}{\frac{4}{3} π} \cdot 8000$

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.

$\frac{12 r^{3}}{12} = \frac{1}{\frac{4}{3} π} \cdot 8000$

Step 3.1.1.8.2

Divide $r^{3}$ by $1$ .

$r^{3} = \frac{1}{\frac{4}{3} π} \cdot 8000$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\frac{1}{\frac{4}{3} π} \cdot 8000$ .

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Step 3.2.1.1

Combine $\frac{4}{3}$ and $π$ .

$r^{3} = \frac{1}{\frac{4 π}{3}} \cdot 8000$

Step 3.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$r^{3} = 1 \frac{3}{4 π} \cdot 8000$

Step 3.2.1.3

Multiply $\frac{3}{4 π}$ by $1$ .

$r^{3} = \frac{3}{4 π} \cdot 8000$

Step 3.2.1.4

Cancel the common factor of $4$ .

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Step 3.2.1.4.1

Factor $4$ out of $4 π$ .

$r^{3} = \frac{3}{4 (π)} \cdot 8000$

Step 3.2.1.4.2

Factor $4$ out of $8000$ .

$r^{3} = \frac{3}{4 (π)} \cdot (4 (2000))$

Step 3.2.1.4.3

Cancel the common factor.

$r^{3} = \frac{3}{4 π} \cdot (4 \cdot 2000)$

Step 3.2.1.4.4

Rewrite the expression.

$r^{3} = \frac{3}{π} \cdot 2000$

Step 3.2.1.5

Combine $\frac{3}{π}$ and $2000$ .

$r^{3} = \frac{3 \cdot 2000}{π}$

Step 3.2.1.6

Multiply $3$ by $2000$ .

$r^{3} = \frac{6000}{π}$

Step 4

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

$r = \sqrt[3]{\frac{6000}{π}}$

Step 5

Simplify $\sqrt[3]{\frac{6000}{π}}$ .

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Step 5.1

Rewrite $\sqrt[3]{\frac{6000}{π}}$ as $\frac{\sqrt[3]{6000}}{\sqrt[3]{π}}$ .

$r = \frac{\sqrt[3]{6000}}{\sqrt[3]{π}}$

Step 5.2

Simplify the numerator.

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Step 5.2.1

Rewrite $6000$ as $10^{3} \cdot 6$ .

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Step 5.2.1.1

Factor $1000$ out of $6000$ .

$r = \frac{\sqrt[3]{1000 (6)}}{\sqrt[3]{π}}$

Step 5.2.1.2

Rewrite $1000$ as $10^{3}$ .

$r = \frac{\sqrt[3]{10^{3} \cdot 6}}{\sqrt[3]{π}}$

Step 5.2.2

Pull terms out from under the radical.

$r = \frac{10 \sqrt[3]{6}}{\sqrt[3]{π}}$

Step 5.3

Multiply $\frac{10 \sqrt[3]{6}}{\sqrt[3]{π}}$ by $\frac{{\sqrt[3]{π}}^{2}}{{\sqrt[3]{π}}^{2}}$ .

$r = \frac{10 \sqrt[3]{6}}{\sqrt[3]{π}} \cdot \frac{{\sqrt[3]{π}}^{2}}{{\sqrt[3]{π}}^{2}}$

Step 5.4

Combine and simplify the denominator.

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Step 5.4.1

Multiply $\frac{10 \sqrt[3]{6}}{\sqrt[3]{π}}$ by $\frac{{\sqrt[3]{π}}^{2}}{{\sqrt[3]{π}}^{2}}$ .

$r = \frac{10 \sqrt[3]{6} {\sqrt[3]{π}}^{2}}{\sqrt[3]{π} {\sqrt[3]{π}}^{2}}$

Step 5.4.2

Raise $\sqrt[3]{π}$ to the power of $1$ .

$r = \frac{10 \sqrt[3]{6} {\sqrt[3]{π}}^{2}}{{\sqrt[3]{π}}^{1} {\sqrt[3]{π}}^{2}}$

Step 5.4.3

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

$r = \frac{10 \sqrt[3]{6} {\sqrt[3]{π}}^{2}}{{\sqrt[3]{π}}^{1 + 2}}$

Step 5.4.4

Add $1$ and $2$ .

$r = \frac{10 \sqrt[3]{6} {\sqrt[3]{π}}^{2}}{{\sqrt[3]{π}}^{3}}$

Step 5.4.5

Rewrite ${\sqrt[3]{π}}^{3}$ as $π$ .

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Step 5.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{π}$ as $π^{\frac{1}{3}}$ .

$r = \frac{10 \sqrt[3]{6} {\sqrt[3]{π}}^{2}}{{(π^{\frac{1}{3}})}^{3}}$

Step 5.4.5.2

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

$r = \frac{10 \sqrt[3]{6} {\sqrt[3]{π}}^{2}}{π^{\frac{1}{3} \cdot 3}}$

Step 5.4.5.3

Combine $\frac{1}{3}$ and $3$ .

$r = \frac{10 \sqrt[3]{6} {\sqrt[3]{π}}^{2}}{π^{\frac{3}{3}}}$

Step 5.4.5.4

Cancel the common factor of $3$ .

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Step 5.4.5.4.1

Cancel the common factor.

$r = \frac{10 \sqrt[3]{6} {\sqrt[3]{π}}^{2}}{π^{\frac{3}{3}}}$

Step 5.4.5.4.2

Rewrite the expression.

$r = \frac{10 \sqrt[3]{6} {\sqrt[3]{π}}^{2}}{π^{1}}$

Step 5.4.5.5

Simplify.

$r = \frac{10 \sqrt[3]{6} {\sqrt[3]{π}}^{2}}{π}$

Step 5.5

Simplify the numerator.

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Step 5.5.1

Rewrite ${\sqrt[3]{π}}^{2}$ as $\sqrt[3]{π^{2}}$ .

$r = \frac{10 \sqrt[3]{6} \sqrt[3]{π^{2}}}{π}$

Step 5.5.2

Combine using the product rule for radicals.

$r = \frac{10 \sqrt[3]{π^{2} \cdot 6}}{π}$

Step 5.6

Move $6$ to the left of $π^{2}$ .

$r = \frac{10 \sqrt[3]{6 π^{2}}}{π}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$r = \frac{10 \sqrt[3]{6 π^{2}}}{π}$

Decimal Form:

$r = 12.40700981 \dots$