Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Apply the product rule to $x^{2} y^{3}$ .

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

Step 1.2

Multiply the exponents in ${(x^{2})}^{\frac{1}{3}}$ .

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Step 1.2.1

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

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

Step 1.2.2

Combine $2$ and $\frac{1}{3}$ .

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

Step 1.3

Multiply the exponents in ${(y^{3})}^{\frac{1}{3}}$ .

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Step 1.3.1

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

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

Step 1.3.2

Cancel the common factor of $3$ .

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Step 1.3.2.1

Cancel the common factor.

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

Step 1.3.2.2

Rewrite the expression.

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

Step 1.4

Simplify.

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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Step 3.1

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

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

Step 3.2

Raise $\sqrt[3]{x^{2} y}$ to the power of $1$ .

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

Step 3.3

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

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

Step 3.4

Add $1$ and $2$ .

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

Step 3.5

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

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Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2} y}$ as ${(x^{2} y)}^{\frac{1}{3}}$ .

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

Step 3.5.2

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

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

Step 3.5.3

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

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

Step 3.5.4

Cancel the common factor of $3$ .

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Step 3.5.4.1

Cancel the common factor.

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

Step 3.5.4.2

Rewrite the expression.

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

Step 3.5.5

Simplify.

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

Step 4

Move $x^{\frac{2}{3}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 5

Multiply $x^{2}$ by $x^{- \frac{2}{3}}$ by adding the exponents.

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Step 5.1

Move $x^{- \frac{2}{3}}$ .

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

Step 5.2

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

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

Step 5.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 5.4

Combine $2$ and $\frac{3}{3}$ .

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

Step 5.5

Combine the numerators over the common denominator.

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

Step 5.6

Simplify the numerator.

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Step 5.6.1

Multiply $2$ by $3$ .

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

Step 5.6.2

Add $- 2$ and $6$ .

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

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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

Step 7

Simplify the numerator.

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Step 7.1

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

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

Step 7.2

Apply the product rule to $x^{2} y$ .

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

Step 7.3

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

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Step 7.3.1

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

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

Step 7.3.2

Multiply $2$ by $2$ .

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

Step 7.4

Rewrite $x^{4} y^{2}$ as $x^{3} (x y^{2})$ .

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Step 7.4.1

Factor out $x^{3}$ .

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

Step 7.4.2

Add parentheses.

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

Step 7.5

Pull terms out from under the radical.

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

Step 8

Move $x^{1}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{\sqrt[3]{x y^{2}}}{x^{\frac{4}{3}} x^{- 1}}$

Step 9

Multiply $x^{\frac{4}{3}}$ by $x^{- 1}$ by adding the exponents.

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Step 9.1

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

$\frac{\sqrt[3]{x y^{2}}}{x^{\frac{4}{3} - 1}}$

Step 9.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\sqrt[3]{x y^{2}}}{x^{\frac{4}{3} - 1 \cdot \frac{3}{3}}}$

Step 9.3

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

$\frac{\sqrt[3]{x y^{2}}}{x^{\frac{4}{3} + \frac{- 1 \cdot 3}{3}}}$

Step 9.4

Combine the numerators over the common denominator.

$\frac{\sqrt[3]{x y^{2}}}{x^{\frac{4 - 1 \cdot 3}{3}}}$

Step 9.5

Simplify the numerator.

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Step 9.5.1

Multiply $- 1$ by $3$ .

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

Step 9.5.2

Subtract $3$ from $4$ .

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