Algebra Examples

$\frac{4 x^{- \frac{1}{2}} y^{\frac{7}{4}}}{\sqrt[4]{16 x^{3} y^{5}}}$

Step 1

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

$\frac{4 y^{\frac{7}{4}}}{\sqrt[4]{16 x^{3} y^{5}} x^{\frac{1}{2}}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $16 x^{3} y^{5}$ as ${(2 y)}^{4} (x^{3} y)$ .

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Step 2.1.1

Rewrite $16$ as $2^{4}$ .

$\frac{4 y^{\frac{7}{4}}}{\sqrt[4]{2^{4} x^{3} y^{5}} x^{\frac{1}{2}}}$

Step 2.1.2

Factor out $y^{4}$ .

$\frac{4 y^{\frac{7}{4}}}{\sqrt[4]{2^{4} x^{3} (y^{4} y)} x^{\frac{1}{2}}}$

Step 2.1.3

Move $x^{3}$ .

$\frac{4 y^{\frac{7}{4}}}{\sqrt[4]{2^{4} y^{4} x^{3} y} x^{\frac{1}{2}}}$

Step 2.1.4

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

$\frac{4 y^{\frac{7}{4}}}{\sqrt[4]{{(2 y)}^{4} x^{3} y} x^{\frac{1}{2}}}$

Step 2.1.5

Add parentheses.

$\frac{4 y^{\frac{7}{4}}}{\sqrt[4]{{(2 y)}^{4} (x^{3} y)} x^{\frac{1}{2}}}$

Step 2.2

Pull terms out from under the radical.

$\frac{4 y^{\frac{7}{4}}}{2 y \sqrt[4]{x^{3} y} x^{\frac{1}{2}}}$

Step 3

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

$\frac{4 y^{\frac{7}{4}} y^{- 1}}{2 \sqrt[4]{x^{3} y} x^{\frac{1}{2}}}$

Step 4

Multiply $y^{\frac{7}{4}}$ by $y^{- 1}$ by adding the exponents.

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Step 4.1

Move $y^{- 1}$ .

$\frac{4 (y^{- 1} y^{\frac{7}{4}})}{2 \sqrt[4]{x^{3} y} x^{\frac{1}{2}}}$

Step 4.2

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

$\frac{4 y^{- 1 + \frac{7}{4}}}{2 \sqrt[4]{x^{3} y} x^{\frac{1}{2}}}$

Step 4.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{4 y^{- 1 \cdot \frac{4}{4} + \frac{7}{4}}}{2 \sqrt[4]{x^{3} y} x^{\frac{1}{2}}}$

Step 4.4

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

$\frac{4 y^{\frac{- 1 \cdot 4}{4} + \frac{7}{4}}}{2 \sqrt[4]{x^{3} y} x^{\frac{1}{2}}}$

Step 4.5

Combine the numerators over the common denominator.

$\frac{4 y^{\frac{- 1 \cdot 4 + 7}{4}}}{2 \sqrt[4]{x^{3} y} x^{\frac{1}{2}}}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $4$ .

$\frac{4 y^{\frac{- 4 + 7}{4}}}{2 \sqrt[4]{x^{3} y} x^{\frac{1}{2}}}$

Step 4.6.2

Add $- 4$ and $7$ .

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

Step 5

Factor $2$ out of $4 y^{\frac{3}{4}}$ .

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

Step 6

Cancel the common factors.

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Step 6.1

Factor $2$ out of $2 \sqrt[4]{x^{3} y} x^{\frac{1}{2}}$ .

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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

Step 7

Multiply $\frac{2 y^{\frac{3}{4}}}{\sqrt[4]{x^{3} y} x^{\frac{1}{2}}}$ by $\frac{{\sqrt[4]{x^{3} y}}^{3}}{{\sqrt[4]{x^{3} y}}^{3}}$ .

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

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply $\frac{2 y^{\frac{3}{4}}}{\sqrt[4]{x^{3} y} x^{\frac{1}{2}}}$ by $\frac{{\sqrt[4]{x^{3} y}}^{3}}{{\sqrt[4]{x^{3} y}}^{3}}$ .

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

Step 8.2

Move $\sqrt[4]{x^{3} y}$ .

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

Step 8.3

Raise $\sqrt[4]{x^{3} y}$ to the power of $1$ .

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

Step 8.4

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

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

Step 8.5

Add $1$ and $3$ .

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

Step 8.6

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

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Step 8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x^{3} y}$ as ${(x^{3} y)}^{\frac{1}{4}}$ .

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

Step 8.6.2

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

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

Step 8.6.3

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

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

Step 8.6.4

Cancel the common factor of $4$ .

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Step 8.6.4.1

Cancel the common factor.

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

Step 8.6.4.2

Rewrite the expression.

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

Step 8.6.5

Simplify.

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

Step 9

Multiply $x^{\frac{1}{2}}$ by $x^{3}$ by adding the exponents.

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Step 9.1

Move $x^{3}$ .

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

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

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

Step 9.5

Combine the numerators over the common denominator.

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

Step 9.6

Simplify the numerator.

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Step 9.6.1

Multiply $3$ by $2$ .

$\frac{2 y^{\frac{3}{4}} {\sqrt[4]{x^{3} y}}^{3}}{x^{\frac{6 + 1}{2}} y}$

Step 9.6.2

Add $6$ and $1$ .

$\frac{2 y^{\frac{3}{4}} {\sqrt[4]{x^{3} y}}^{3}}{x^{\frac{7}{2}} y}$

Step 10

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

$\frac{2 {\sqrt[4]{x^{3} y}}^{3}}{x^{\frac{7}{2}} y \cdot y^{- \frac{3}{4}}}$

Step 11

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

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Step 11.1

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

$\frac{2 {\sqrt[4]{x^{3} y}}^{3}}{x^{\frac{7}{2}} (y^{- \frac{3}{4}} y)}$

Step 11.2

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

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Step 11.2.1

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

$\frac{2 {\sqrt[4]{x^{3} y}}^{3}}{x^{\frac{7}{2}} (y^{- \frac{3}{4}} y^{1})}$

Step 11.2.2

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

$\frac{2 {\sqrt[4]{x^{3} y}}^{3}}{x^{\frac{7}{2}} y^{- \frac{3}{4} + 1}}$

Step 11.3

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

$\frac{2 {\sqrt[4]{x^{3} y}}^{3}}{x^{\frac{7}{2}} y^{- \frac{3}{4} + \frac{4}{4}}}$

Step 11.4

Combine the numerators over the common denominator.

$\frac{2 {\sqrt[4]{x^{3} y}}^{3}}{x^{\frac{7}{2}} y^{\frac{- 3 + 4}{4}}}$

Step 11.5

Add $- 3$ and $4$ .

$\frac{2 {\sqrt[4]{x^{3} y}}^{3}}{x^{\frac{7}{2}} y^{\frac{1}{4}}}$

Step 12

Simplify the numerator.

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Step 12.1

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

$\frac{2 \sqrt[4]{{(x^{3} y)}^{3}}}{x^{\frac{7}{2}} y^{\frac{1}{4}}}$

Step 12.2

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

$\frac{2 \sqrt[4]{{(x^{3})}^{3} y^{3}}}{x^{\frac{7}{2}} y^{\frac{1}{4}}}$

Step 12.3

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

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Step 12.3.1

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

$\frac{2 \sqrt[4]{x^{3 \cdot 3} y^{3}}}{x^{\frac{7}{2}} y^{\frac{1}{4}}}$

Step 12.3.2

Multiply $3$ by $3$ .

$\frac{2 \sqrt[4]{x^{9} y^{3}}}{x^{\frac{7}{2}} y^{\frac{1}{4}}}$

Step 12.4

Rewrite $x^{9} y^{3}$ as ${(x^{2})}^{4} (x y^{3})$ .

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Step 12.4.1

Factor out $x^{8}$ .

$\frac{2 \sqrt[4]{x^{8} x y^{3}}}{x^{\frac{7}{2}} y^{\frac{1}{4}}}$

Step 12.4.2

Rewrite $x^{8}$ as ${(x^{2})}^{4}$ .

$\frac{2 \sqrt[4]{{(x^{2})}^{4} x y^{3}}}{x^{\frac{7}{2}} y^{\frac{1}{4}}}$

Step 12.4.3

Add parentheses.

$\frac{2 \sqrt[4]{{(x^{2})}^{4} (x y^{3})}}{x^{\frac{7}{2}} y^{\frac{1}{4}}}$

Step 12.5

Pull terms out from under the radical.

$\frac{2 x^{2} \sqrt[4]{x y^{3}}}{x^{\frac{7}{2}} y^{\frac{1}{4}}}$

Step 13

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

$\frac{2 \sqrt[4]{x y^{3}}}{x^{\frac{7}{2}} y^{\frac{1}{4}} x^{- 2}}$

Step 14

Multiply $x^{\frac{7}{2}}$ by $x^{- 2}$ by adding the exponents.

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Step 14.1

Move $x^{- 2}$ .

$\frac{2 \sqrt[4]{x y^{3}}}{x^{- 2} x^{\frac{7}{2}} y^{\frac{1}{4}}}$

Step 14.2

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

$\frac{2 \sqrt[4]{x y^{3}}}{x^{- 2 + \frac{7}{2}} y^{\frac{1}{4}}}$

Step 14.3

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

$\frac{2 \sqrt[4]{x y^{3}}}{x^{- 2 \cdot \frac{2}{2} + \frac{7}{2}} y^{\frac{1}{4}}}$

Step 14.4

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

$\frac{2 \sqrt[4]{x y^{3}}}{x^{\frac{- 2 \cdot 2}{2} + \frac{7}{2}} y^{\frac{1}{4}}}$

Step 14.5

Combine the numerators over the common denominator.

$\frac{2 \sqrt[4]{x y^{3}}}{x^{\frac{- 2 \cdot 2 + 7}{2}} y^{\frac{1}{4}}}$

Step 14.6

Simplify the numerator.

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Step 14.6.1

Multiply $- 2$ by $2$ .

$\frac{2 \sqrt[4]{x y^{3}}}{x^{\frac{- 4 + 7}{2}} y^{\frac{1}{4}}}$

Step 14.6.2

Add $- 4$ and $7$ .

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