Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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Step 1.1.1

Factor out $y^{2}$ .

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

Step 1.1.2

Reorder $x$ and $y^{2}$ .

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

Step 1.1.3

Add parentheses.

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

Combine and simplify the denominator.

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.1.4

Add $1$ and $3$ .

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

Step 3.1.5

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

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Step 3.1.5.1

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

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

Step 3.1.5.2

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

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

Step 3.1.5.3

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

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

Step 3.1.5.4

Cancel the common factor of $4$ .

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Step 3.1.5.4.1

Cancel the common factor.

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

Step 3.1.5.4.2

Rewrite the expression.

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

Step 3.1.5.5

Simplify.

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

Step 3.2

Cancel the common factor of $y$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

Multiply $3$ by $3$ .

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

Step 4.4

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

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Step 4.4.1

Factor out $x^{8}$ .

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

Step 4.4.2

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

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

Step 4.4.3

Add parentheses.

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

Step 4.5

Pull terms out from under the radical.

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

Step 4.6

Combine exponents.

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Step 4.6.1

Rewrite the expression using the least common index of $4$ .

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Step 4.6.1.1

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

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

Step 4.6.1.2

Rewrite ${(x y)}^{\frac{1}{2}}$ as ${(x y)}^{\frac{2}{4}}$ .

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

Step 4.6.1.3

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

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

Step 4.6.2

Combine using the product rule for radicals.

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

Step 4.6.3

Apply the product rule to $x y$ .

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

Step 4.6.4

Combine exponents.

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Step 4.6.4.1

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

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

Step 4.6.4.2

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

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

Step 4.6.4.3

Add $1$ and $2$ .

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

Step 4.6.4.4

Multiply $y^{2}$ by $y^{3}$ by adding the exponents.

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Step 4.6.4.4.1

Move $y^{3}$ .

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

Step 4.6.4.4.2

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

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

Step 4.6.4.4.3

Add $3$ and $2$ .

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

Step 4.7

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

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Step 4.7.1

Factor out $y^{4}$ .

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

Step 4.7.2

Reorder $x^{3}$ and $y^{4}$ .

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

Step 4.7.3

Add parentheses.

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

Step 4.8

Pull terms out from under the radical.

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

Step 5

Simplify the numerator.

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Step 5.1

Rewrite.

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

Step 5.2

Rewrite.

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

Step 5.3

Multiply $\sqrt{x y}$ by $1$ .

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

Step 5.4

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

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

Step 5.5

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

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

Step 5.6

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

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Step 5.6.1

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

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

Step 5.6.2

Multiply $3$ by $3$ .

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

Step 5.7

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

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Step 5.7.1

Factor out $x^{8}$ .

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

Step 5.7.2

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

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

Step 5.7.3

Add parentheses.

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

Step 5.8

Pull terms out from under the radical.

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

Step 5.9

Combine exponents.

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Step 5.9.1

Rewrite the expression using the least common index of $4$ .

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Step 5.9.1.1

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

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

Step 5.9.1.2

Rewrite ${(x y)}^{\frac{1}{2}}$ as ${(x y)}^{\frac{2}{4}}$ .

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

Step 5.9.1.3

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

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

Step 5.9.2

Combine using the product rule for radicals.

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

Step 5.9.3

Apply the product rule to $x y$ .

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

Step 5.9.4

Combine exponents.

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Step 5.9.4.1

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

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

Step 5.9.4.2

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

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

Step 5.9.4.3

Add $1$ and $2$ .

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

Step 5.9.4.4

Multiply $y^{2}$ by $y^{3}$ by adding the exponents.

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Step 5.9.4.4.1

Move $y^{3}$ .

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

Step 5.9.4.4.2

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

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

Step 5.9.4.4.3

Add $3$ and $2$ .

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

Step 5.10

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

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Step 5.10.1

Factor out $y^{4}$ .

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

Step 5.10.2

Reorder $x^{3}$ and $y^{4}$ .

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

Step 5.10.3

Add parentheses.

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

Step 5.11

Pull terms out from under the radical.

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

Step 5.12

Remove unnecessary parentheses.

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

Step 6

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 6.1

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

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

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $x^{2}$ out of $x^{3}$ .

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

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