Pre-Algebra Examples

$\sqrt[3]{\frac{- 432 x^{9}}{125 y^{7}}}$

Step 1

Move the negative in front of the fraction.

$\sqrt[3]{- \frac{432 x^{9}}{125 y^{7}}}$

Step 2

Rewrite $- \frac{432 x^{9}}{125 y^{7}}$ as ${(- \frac{6 x^{3}}{5 y^{2}})}^{3} \frac{2}{y}$ .

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Step 2.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$\sqrt[3]{{(- 1)}^{3} \frac{432 x^{9}}{125 y^{7}}}$

Step 2.2

Factor the perfect power ${(6 x^{3})}^{3}$ out of $432 x^{9}$ .

$\sqrt[3]{{(- 1)}^{3} \frac{{(6 x^{3})}^{3} \cdot 2}{125 y^{7}}}$

Step 2.3

Factor the perfect power ${(5 y^{2})}^{3}$ out of $125 y^{7}$ .

$\sqrt[3]{{(- 1)}^{3} \frac{{(6 x^{3})}^{3} \cdot 2}{{(5 y^{2})}^{3} y}}$

Step 2.4

Rearrange the fraction $\frac{{(6 x^{3})}^{3} \cdot 2}{{(5 y^{2})}^{3} y}$ .

$\sqrt[3]{{(- 1)}^{3} ({(\frac{6 x^{3}}{5 y^{2}})}^{3} \frac{2}{y})}$

Step 2.5

Rewrite ${(- 1)}^{3} {(\frac{6 x^{3}}{5 y^{2}})}^{3}$ as ${(- \frac{6 x^{3}}{5 y^{2}})}^{3}$ .

$\sqrt[3]{{(- \frac{6 x^{3}}{5 y^{2}})}^{3} \frac{2}{y}}$

Step 3

Pull terms out from under the radical.

$- \frac{6 x^{3}}{5 y^{2}} \sqrt[3]{\frac{2}{y}}$

Step 4

Rewrite $\sqrt[3]{\frac{2}{y}}$ as $\frac{\sqrt[3]{2}}{\sqrt[3]{y}}$ .

$- \frac{6 x^{3}}{5 y^{2}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{y}}$

Step 5

Multiply $\frac{\sqrt[3]{2}}{\sqrt[3]{y}}$ by $\frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{2}}$ .

$- \frac{6 x^{3}}{5 y^{2}} (\frac{\sqrt[3]{2}}{\sqrt[3]{y}} \cdot \frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{2}})$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{\sqrt[3]{2}}{\sqrt[3]{y}}$ by $\frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{2}}$ .

$- \frac{6 x^{3}}{5 y^{2}} \cdot \frac{\sqrt[3]{2} {\sqrt[3]{y}}^{2}}{\sqrt[3]{y} {\sqrt[3]{y}}^{2}}$

Step 6.2

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

$- \frac{6 x^{3}}{5 y^{2}} \cdot \frac{\sqrt[3]{2} {\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{1} {\sqrt[3]{y}}^{2}}$

Step 6.3

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

$- \frac{6 x^{3}}{5 y^{2}} \cdot \frac{\sqrt[3]{2} {\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{1 + 2}}$

Step 6.4

Add $1$ and $2$ .

$- \frac{6 x^{3}}{5 y^{2}} \cdot \frac{\sqrt[3]{2} {\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{3}}$

Step 6.5

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

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Step 6.5.1

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

$- \frac{6 x^{3}}{5 y^{2}} \cdot \frac{\sqrt[3]{2} {\sqrt[3]{y}}^{2}}{{(y^{\frac{1}{3}})}^{3}}$

Step 6.5.2

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

$- \frac{6 x^{3}}{5 y^{2}} \cdot \frac{\sqrt[3]{2} {\sqrt[3]{y}}^{2}}{y^{\frac{1}{3} \cdot 3}}$

Step 6.5.3

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

$- \frac{6 x^{3}}{5 y^{2}} \cdot \frac{\sqrt[3]{2} {\sqrt[3]{y}}^{2}}{y^{\frac{3}{3}}}$

Step 6.5.4

Cancel the common factor of $3$ .

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Step 6.5.4.1

Cancel the common factor.

$- \frac{6 x^{3}}{5 y^{2}} \cdot \frac{\sqrt[3]{2} {\sqrt[3]{y}}^{2}}{y^{\frac{3}{3}}}$

Step 6.5.4.2

Rewrite the expression.

$- \frac{6 x^{3}}{5 y^{2}} \cdot \frac{\sqrt[3]{2} {\sqrt[3]{y}}^{2}}{y^{1}}$

Step 6.5.5

Simplify.

$- \frac{6 x^{3}}{5 y^{2}} \cdot \frac{\sqrt[3]{2} {\sqrt[3]{y}}^{2}}{y}$

Step 7

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

$- \frac{6 x^{3}}{5 y^{2}} \cdot \frac{\sqrt[3]{2} \sqrt[3]{y^{2}}}{y}$

Step 8

Combine using the product rule for radicals.

$- \frac{6 x^{3}}{5 y^{2}} \cdot \frac{\sqrt[3]{2 y^{2}}}{y}$

Step 9

Multiply $- \frac{6 x^{3}}{5 y^{2}} \cdot \frac{\sqrt[3]{2 y^{2}}}{y}$ .

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Step 9.1

Multiply $\frac{\sqrt[3]{2 y^{2}}}{y}$ by $\frac{6 x^{3}}{5 y^{2}}$ .

$- \frac{\sqrt[3]{2 y^{2}} (6 x^{3})}{y (5 y^{2})}$

Step 9.2

Raise $y$ to the power of $1$ .

$- \frac{\sqrt[3]{2 y^{2}} (6 x^{3})}{5 (y^{1} y^{2})}$

Step 9.3

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

$- \frac{\sqrt[3]{2 y^{2}} (6 x^{3})}{5 y^{1 + 2}}$

Step 9.4

Add $1$ and $2$ .

$- \frac{\sqrt[3]{2 y^{2}} (6 x^{3})}{5 y^{3}}$

Step 10

Move $6$ to the left of $\sqrt[3]{2 y^{2}}$ .

$- \frac{6 \sqrt[3]{2 y^{2}} x^{3}}{5 y^{3}}$