Pre-Algebra Examples

$\frac{\sqrt[3]{x}}{\sqrt[3]{9 y}}$

Step 1

Multiply $\frac{\sqrt[3]{x}}{\sqrt[3]{9 y}}$ by $\frac{{\sqrt[3]{9 y}}^{2}}{{\sqrt[3]{9 y}}^{2}}$ .

$\frac{\sqrt[3]{x}}{\sqrt[3]{9 y}} \cdot \frac{{\sqrt[3]{9 y}}^{2}}{{\sqrt[3]{9 y}}^{2}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{\sqrt[3]{x}}{\sqrt[3]{9 y}}$ by $\frac{{\sqrt[3]{9 y}}^{2}}{{\sqrt[3]{9 y}}^{2}}$ .

$\frac{\sqrt[3]{x} {\sqrt[3]{9 y}}^{2}}{\sqrt[3]{9 y} {\sqrt[3]{9 y}}^{2}}$

Step 2.2

Raise $\sqrt[3]{9 y}$ to the power of $1$ .

$\frac{\sqrt[3]{x} {\sqrt[3]{9 y}}^{2}}{{\sqrt[3]{9 y}}^{1} {\sqrt[3]{9 y}}^{2}}$

Step 2.3

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

$\frac{\sqrt[3]{x} {\sqrt[3]{9 y}}^{2}}{{\sqrt[3]{9 y}}^{1 + 2}}$

Step 2.4

Add $1$ and $2$ .

$\frac{\sqrt[3]{x} {\sqrt[3]{9 y}}^{2}}{{\sqrt[3]{9 y}}^{3}}$

Step 2.5

Rewrite ${\sqrt[3]{9 y}}^{3}$ as $9 y$ .

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{9 y}$ as ${(9 y)}^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{x} {\sqrt[3]{9 y}}^{2}}{{({(9 y)}^{\frac{1}{3}})}^{3}}$

Step 2.5.2

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

$\frac{\sqrt[3]{x} {\sqrt[3]{9 y}}^{2}}{{(9 y)}^{\frac{1}{3} \cdot 3}}$

Step 2.5.3

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

$\frac{\sqrt[3]{x} {\sqrt[3]{9 y}}^{2}}{{(9 y)}^{\frac{3}{3}}}$

Step 2.5.4

Cancel the common factor of $3$ .

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Step 2.5.4.1

Cancel the common factor.

$\frac{\sqrt[3]{x} {\sqrt[3]{9 y}}^{2}}{{(9 y)}^{\frac{3}{3}}}$

Step 2.5.4.2

Rewrite the expression.

$\frac{\sqrt[3]{x} {\sqrt[3]{9 y}}^{2}}{{(9 y)}^{1}}$

Step 2.5.5

Simplify.

$\frac{\sqrt[3]{x} {\sqrt[3]{9 y}}^{2}}{9 y}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite ${\sqrt[3]{9 y}}^{2}$ as $\sqrt[3]{{(9 y)}^{2}}$ .

$\frac{\sqrt[3]{x} \sqrt[3]{{(9 y)}^{2}}}{9 y}$

Step 3.2

Apply the product rule to $9 y$ .

$\frac{\sqrt[3]{x} \sqrt[3]{9^{2} y^{2}}}{9 y}$

Step 3.3

Raise $9$ to the power of $2$ .

$\frac{\sqrt[3]{x} \sqrt[3]{81 y^{2}}}{9 y}$

Step 3.4

Rewrite $81 y^{2}$ as $3^{3} \cdot (3 y^{2})$ .

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Step 3.4.1

Factor $27$ out of $81$ .

$\frac{\sqrt[3]{x} \sqrt[3]{27 (3) y^{2}}}{9 y}$

Step 3.4.2

Rewrite $27$ as $3^{3}$ .

$\frac{\sqrt[3]{x} \sqrt[3]{3^{3} \cdot 3 y^{2}}}{9 y}$

Step 3.4.3

Add parentheses.

$\frac{\sqrt[3]{x} \sqrt[3]{3^{3} \cdot (3 y^{2})}}{9 y}$

Step 3.5

Pull terms out from under the radical.

$\frac{\sqrt[3]{x} \cdot 3 \sqrt[3]{3 y^{2}}}{9 y}$

Step 3.6

Combine using the product rule for radicals.

$\frac{3 \sqrt[3]{x \cdot 3 y^{2}}}{9 y}$

Step 4

Cancel the common factor of $3$ and $9$ .

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Step 4.1

Factor $3$ out of $3 \sqrt[3]{x \cdot 3 y^{2}}$ .

$\frac{3 (\sqrt[3]{x \cdot 3 y^{2}})}{9 y}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $3$ out of $9 y$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 5

Reorder factors in $\frac{\sqrt[3]{x \cdot 3 y^{2}}}{3 y}$ .

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