Algebra Examples

$\sqrt[3]{x^{5} y}$

Step 1

Rewrite $\sqrt[3]{x^{5} y}$ as $\sqrt[3]{x^{5}} \sqrt[3]{y}$ .

$\sqrt[3]{x^{5}} \sqrt[3]{y}$

Step 2

Multiply to rationalize the numerator.

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

Step 3

Simplify.

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Step 3.1

Raise $\sqrt[3]{x^{5}}$ to the power of $1$ .

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

Step 3.2

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

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

Step 3.3

Add $1$ and $2$ .

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

Step 3.4

Rewrite ${\sqrt[3]{x^{5}}}^{3}$ as $x^{5}$ .

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Step 3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{5}}$ as $x^{\frac{5}{3}}$ .

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

Step 3.4.2

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

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

Step 3.4.3

Combine $\frac{5}{3}$ and $3$ .

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

Step 3.4.4

Multiply $5$ by $3$ .

$\frac{\sqrt[3]{y} x^{\frac{15}{3}}}{{\sqrt[3]{x^{5}}}^{2}}$

Step 3.4.5

Cancel the common factor of $15$ and $3$ .

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Step 3.4.5.1

Factor $3$ out of $15$ .

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

Step 3.4.5.2

Cancel the common factors.

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Step 3.4.5.2.1

Factor $3$ out of $3$ .

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

Step 3.4.5.2.2

Cancel the common factor.

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

Step 3.4.5.2.3

Rewrite the expression.

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

Step 3.4.5.2.4

Divide $5$ by $1$ .

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

Step 4

Multiply to rationalize the numerator.

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Add $1$ and $2$ .

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

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

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

Step 5.4.3

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

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

Step 5.4.4

Cancel the common factor of $3$ .

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Step 5.4.4.1

Cancel the common factor.

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

Step 5.4.4.2

Rewrite the expression.

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

Step 5.4.5

Simplify.

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

Step 5.5

Start simplifying.

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

Step 5.6

Factor out $x^{3}$ .

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

Step 5.7

Pull terms out from under the radical.

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

Step 5.8

Apply the product rule to $x \sqrt[3]{x^{2}}$ .

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

Step 5.9

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

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

Step 5.10

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 5.10.1

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

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

Step 5.10.2

Multiply $2$ by $2$ .

$\frac{x^{5} y}{x^{2} \sqrt[3]{x^{4}} \sqrt[3]{y^{2}}}$

Step 5.11

Factor out $x^{3}$ .

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

Step 5.12

Pull terms out from under the radical.

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

Step 5.13

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 5.13.1

Move $x$ .

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

Step 5.13.2

Multiply $x$ by $x^{2}$ .

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Step 5.13.2.1

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

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

Step 5.13.2.2

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

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

Step 5.13.3

Add $1$ and $2$ .

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

Step 5.14

Combine using the product rule for radicals.

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