Algebra Examples

$\frac{x^{4} y^{7}}{\sqrt[3]{x^{10} y^{4}}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $x^{10} y^{4}$ as ${(x^{3} y)}^{3} (x y)$ .

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Step 1.1.1

Factor out $x^{9}$ .

$\frac{x^{4} y^{7}}{\sqrt[3]{x^{9} x y^{4}}}$

Step 1.1.2

Rewrite $x^{9}$ as ${(x^{3})}^{3}$ .

$\frac{x^{4} y^{7}}{\sqrt[3]{{(x^{3})}^{3} x y^{4}}}$

Step 1.1.3

Factor out $y^{3}$ .

$\frac{x^{4} y^{7}}{\sqrt[3]{{(x^{3})}^{3} x (y^{3} y)}}$

Step 1.1.4

Move $x$ .

$\frac{x^{4} y^{7}}{\sqrt[3]{{(x^{3})}^{3} y^{3} x y}}$

Step 1.1.5

Rewrite ${(x^{3})}^{3} y^{3}$ as ${(x^{3} y)}^{3}$ .

$\frac{x^{4} y^{7}}{\sqrt[3]{{(x^{3} y)}^{3} x y}}$

Step 1.1.6

Add parentheses.

$\frac{x^{4} y^{7}}{\sqrt[3]{{(x^{3} y)}^{3} (x y)}}$

Step 1.2

Pull terms out from under the radical.

$\frac{x^{4} y^{7}}{x^{3} y \sqrt[3]{x y}}$

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $x^{4}$ and $x^{3}$ .

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Step 2.1.1

Factor $x^{3}$ out of $x^{4} y^{7}$ .

$\frac{x^{3} (x y^{7})}{x^{3} y \sqrt[3]{x y}}$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

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

$\frac{x^{3} (x y^{7})}{x^{3} (y \sqrt[3]{x y})}$

Step 2.1.2.2

Cancel the common factor.

$\frac{x^{3} (x y^{7})}{x^{3} (y \sqrt[3]{x y})}$

Step 2.1.2.3

Rewrite the expression.

$\frac{x y^{7}}{y \sqrt[3]{x y}}$

Step 2.2

Cancel the common factor of $y^{7}$ and $y$ .

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Step 2.2.1

Factor $y$ out of $x y^{7}$ .

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

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

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

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

Step 2.2.2.2

Cancel the common factor.

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

Step 2.2.2.3

Rewrite the expression.

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

Step 3

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

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

Step 4

Combine and simplify the denominator.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Add $1$ and $2$ .

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

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

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

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

Step 4.5.3

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

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

Step 4.5.4

Cancel the common factor of $3$ .

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Step 4.5.4.1

Cancel the common factor.

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

Step 4.5.4.2

Rewrite the expression.

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

Step 4.5.5

Simplify.

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

Step 5

Cancel the common factor of $x$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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

Step 6

Cancel the common factor of $y^{6}$ and $y$ .

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Step 6.1

Factor $y$ out of $y^{6} {\sqrt[3]{x y}}^{2}$ .

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

Step 6.2

Cancel the common factors.

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Step 6.2.1

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

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

Step 6.2.2

Factor $y$ out of $y^{1}$ .

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

Step 6.2.3

Cancel the common factor.

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

Step 6.2.4

Rewrite the expression.

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

Step 6.2.5

Divide $y^{5} {\sqrt[3]{x y}}^{2}$ by $1$ .

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

Step 7

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

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

Step 8

Apply the product rule to $x y$ .

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