Algebra Examples

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

Step 1

Multiply to rationalize the numerator.

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Add $1$ and $2$ .

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.4.4

Cancel the common factor of $3$ .

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Step 2.4.4.1

Cancel the common factor.

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

Step 2.4.4.2

Rewrite the expression.

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

Step 2.4.5

Simplify.

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

Step 2.5

Start simplifying.

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

Step 2.6

Apply the product rule to $9 x$ .

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

Step 2.7

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

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

Step 2.8

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

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Step 2.8.1

Factor $27$ out of $81$ .

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

Step 2.8.2

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

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

Step 2.8.3

Add parentheses.

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

Step 2.9

Pull terms out from under the radical.

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

Step 2.10

Rewrite using the commutative property of multiplication.

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

Step 2.11

Combine using the product rule for radicals.

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