Algebra Examples

$\frac{6 x^{\frac{5}{4}} y^{- \frac{5}{4}}}{\sqrt[4]{81 x^{2} y}}$

Step 1

Move $y^{- \frac{5}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{6 x^{\frac{5}{4}}}{\sqrt[4]{81 x^{2} y} y^{\frac{5}{4}}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $81 x^{2} y$ as $3^{4} (x^{2} y)$ .

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Step 2.1.1

Rewrite $81$ as $3^{4}$ .

$\frac{6 x^{\frac{5}{4}}}{\sqrt[4]{3^{4} x^{2} y} y^{\frac{5}{4}}}$

Step 2.1.2

Add parentheses.

$\frac{6 x^{\frac{5}{4}}}{\sqrt[4]{3^{4} (x^{2} y)} y^{\frac{5}{4}}}$

Step 2.2

Pull terms out from under the radical.

$\frac{6 x^{\frac{5}{4}}}{3 \sqrt[4]{x^{2} y} y^{\frac{5}{4}}}$

Step 3

Factor $3$ out of $6 x^{\frac{5}{4}}$ .

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

Step 4

Cancel the common factors.

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Step 4.1

Factor $3$ out of $3 \sqrt[4]{x^{2} y} y^{\frac{5}{4}}$ .

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

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

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

Step 6

Combine and simplify the denominator.

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Step 6.1

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

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

Step 6.2

Move $\sqrt[4]{x^{2} y}$ .

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

Step 6.3

Raise $\sqrt[4]{x^{2} y}$ to the power of $1$ .

$\frac{2 x^{\frac{5}{4}} {\sqrt[4]{x^{2} y}}^{3}}{y^{\frac{5}{4}} ({\sqrt[4]{x^{2} y}}^{1} {\sqrt[4]{x^{2} y}}^{3})}$

Step 6.4

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

$\frac{2 x^{\frac{5}{4}} {\sqrt[4]{x^{2} y}}^{3}}{y^{\frac{5}{4}} {\sqrt[4]{x^{2} y}}^{1 + 3}}$

Step 6.5

Add $1$ and $3$ .

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

Step 6.6

Rewrite ${\sqrt[4]{x^{2} y}}^{4}$ as $x^{2} y$ .

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Step 6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x^{2} y}$ as ${(x^{2} y)}^{\frac{1}{4}}$ .

$\frac{2 x^{\frac{5}{4}} {\sqrt[4]{x^{2} y}}^{3}}{y^{\frac{5}{4}} {({(x^{2} y)}^{\frac{1}{4}})}^{4}}$

Step 6.6.2

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

$\frac{2 x^{\frac{5}{4}} {\sqrt[4]{x^{2} y}}^{3}}{y^{\frac{5}{4}} {(x^{2} y)}^{\frac{1}{4} \cdot 4}}$

Step 6.6.3

Combine $\frac{1}{4}$ and $4$ .

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

Step 6.6.4

Cancel the common factor of $4$ .

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Step 6.6.4.1

Cancel the common factor.

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

Step 6.6.4.2

Rewrite the expression.

$\frac{2 x^{\frac{5}{4}} {\sqrt[4]{x^{2} y}}^{3}}{y^{\frac{5}{4}} {(x^{2} y)}^{1}}$

Step 6.6.5

Simplify.

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

Step 7

Multiply $y^{\frac{5}{4}}$ by $y$ by adding the exponents.

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Step 7.1

Move $y$ .

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

Step 7.2

Multiply $y$ by $y^{\frac{5}{4}}$ .

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Step 7.2.1

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

$\frac{2 x^{\frac{5}{4}} {\sqrt[4]{x^{2} y}}^{3}}{y^{1} y^{\frac{5}{4}} x^{2}}$

Step 7.2.2

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

$\frac{2 x^{\frac{5}{4}} {\sqrt[4]{x^{2} y}}^{3}}{y^{1 + \frac{5}{4}} x^{2}}$

Step 7.3

Write $1$ as a fraction with a common denominator.

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

Step 7.4

Combine the numerators over the common denominator.

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

Step 7.5

Add $4$ and $5$ .

$\frac{2 x^{\frac{5}{4}} {\sqrt[4]{x^{2} y}}^{3}}{y^{\frac{9}{4}} x^{2}}$

Step 8

Move $x^{\frac{5}{4}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{2 {\sqrt[4]{x^{2} y}}^{3}}{y^{\frac{9}{4}} x^{2} x^{- \frac{5}{4}}}$

Step 9

Multiply $x^{2}$ by $x^{- \frac{5}{4}}$ by adding the exponents.

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Step 9.1

Move $x^{- \frac{5}{4}}$ .

$\frac{2 {\sqrt[4]{x^{2} y}}^{3}}{y^{\frac{9}{4}} (x^{- \frac{5}{4}} x^{2})}$

Step 9.2

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

$\frac{2 {\sqrt[4]{x^{2} y}}^{3}}{y^{\frac{9}{4}} x^{- \frac{5}{4} + 2}}$

Step 9.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{2 {\sqrt[4]{x^{2} y}}^{3}}{y^{\frac{9}{4}} x^{- \frac{5}{4} + 2 \cdot \frac{4}{4}}}$

Step 9.4

Combine $2$ and $\frac{4}{4}$ .

$\frac{2 {\sqrt[4]{x^{2} y}}^{3}}{y^{\frac{9}{4}} x^{- \frac{5}{4} + \frac{2 \cdot 4}{4}}}$

Step 9.5

Combine the numerators over the common denominator.

$\frac{2 {\sqrt[4]{x^{2} y}}^{3}}{y^{\frac{9}{4}} x^{\frac{- 5 + 2 \cdot 4}{4}}}$

Step 9.6

Simplify the numerator.

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Step 9.6.1

Multiply $2$ by $4$ .

$\frac{2 {\sqrt[4]{x^{2} y}}^{3}}{y^{\frac{9}{4}} x^{\frac{- 5 + 8}{4}}}$

Step 9.6.2

Add $- 5$ and $8$ .

$\frac{2 {\sqrt[4]{x^{2} y}}^{3}}{y^{\frac{9}{4}} x^{\frac{3}{4}}}$

Step 10

Simplify the numerator.

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Step 10.1

Rewrite ${\sqrt[4]{x^{2} y}}^{3}$ as $\sqrt[4]{{(x^{2} y)}^{3}}$ .

$\frac{2 \sqrt[4]{{(x^{2} y)}^{3}}}{y^{\frac{9}{4}} x^{\frac{3}{4}}}$

Step 10.2

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

$\frac{2 \sqrt[4]{{(x^{2})}^{3} y^{3}}}{y^{\frac{9}{4}} x^{\frac{3}{4}}}$

Step 10.3

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

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Step 10.3.1

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

$\frac{2 \sqrt[4]{x^{2 \cdot 3} y^{3}}}{y^{\frac{9}{4}} x^{\frac{3}{4}}}$

Step 10.3.2

Multiply $2$ by $3$ .

$\frac{2 \sqrt[4]{x^{6} y^{3}}}{y^{\frac{9}{4}} x^{\frac{3}{4}}}$

Step 10.4

Rewrite $x^{6} y^{3}$ as $x^{4} (x^{2} y^{3})$ .

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Step 10.4.1

Factor out $x^{4}$ .

$\frac{2 \sqrt[4]{x^{4} x^{2} y^{3}}}{y^{\frac{9}{4}} x^{\frac{3}{4}}}$

Step 10.4.2

Add parentheses.

$\frac{2 \sqrt[4]{x^{4} (x^{2} y^{3})}}{y^{\frac{9}{4}} x^{\frac{3}{4}}}$

Step 10.5

Pull terms out from under the radical.

$\frac{2 x \sqrt[4]{x^{2} y^{3}}}{y^{\frac{9}{4}} x^{\frac{3}{4}}}$

Step 11

Move $x^{\frac{3}{4}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{2 x \sqrt[4]{x^{2} y^{3}} x^{- \frac{3}{4}}}{y^{\frac{9}{4}}}$

Step 12

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

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Step 12.1

Move $x^{- \frac{3}{4}}$ .

$\frac{2 (x^{- \frac{3}{4}} x) \sqrt[4]{x^{2} y^{3}}}{y^{\frac{9}{4}}}$

Step 12.2

Multiply $x^{- \frac{3}{4}}$ by $x$ .

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Step 12.2.1

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

$\frac{2 (x^{- \frac{3}{4}} x^{1}) \sqrt[4]{x^{2} y^{3}}}{y^{\frac{9}{4}}}$

Step 12.2.2

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

$\frac{2 x^{- \frac{3}{4} + 1} \sqrt[4]{x^{2} y^{3}}}{y^{\frac{9}{4}}}$

Step 12.3

Write $1$ as a fraction with a common denominator.

$\frac{2 x^{- \frac{3}{4} + \frac{4}{4}} \sqrt[4]{x^{2} y^{3}}}{y^{\frac{9}{4}}}$

Step 12.4

Combine the numerators over the common denominator.

$\frac{2 x^{\frac{- 3 + 4}{4}} \sqrt[4]{x^{2} y^{3}}}{y^{\frac{9}{4}}}$

Step 12.5

Add $- 3$ and $4$ .

$\frac{2 x^{\frac{1}{4}} \sqrt[4]{x^{2} y^{3}}}{y^{\frac{9}{4}}}$