Algebra Examples

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

Step 1

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

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

Step 2

Combine and simplify the denominator.

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Step 2.1

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

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

Step 2.2

Raise $\sqrt[4]{x^{3} y^{2}}$ to the power of $1$ .

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

Step 2.3

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

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

Step 2.4

Add $1$ and $3$ .

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

Step 2.5

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

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x^{3} y^{2}}$ as ${(x^{3} y^{2})}^{\frac{1}{4}}$ .

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

Step 2.5.2

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

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

Step 2.5.3

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

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

Step 2.5.4

Cancel the common factor of $4$ .

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Step 2.5.4.1

Cancel the common factor.

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

Step 2.5.4.2

Rewrite the expression.

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

Step 2.5.5

Simplify.

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

Cancel the common factor of $y^{2}$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Multiply $3$ by $3$ .

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

Step 8

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

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Step 8.1

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

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

Step 8.2

Multiply $2$ by $3$ .

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

Step 9

Rewrite $x^{9} y^{6}$ as ${(x^{2} y)}^{4} (x y^{2})$ .

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Step 9.1

Factor out $x^{8}$ .

$x \sqrt[4]{x^{8} x y^{6}}$

Step 9.2

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

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

Step 9.3

Factor out $y^{4}$ .

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

Step 9.4

Move $x$ .

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

Step 9.5

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

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

Step 9.6

Add parentheses.

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

Step 10

Pull terms out from under the radical.

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

Step 11

Remove non-negative terms from the absolute value.

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

Step 12

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 12.1

Move $x^{2}$ .

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

Step 12.2

Multiply $x^{2}$ by $x$ .

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Step 12.2.1

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

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

Step 12.2.2

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

$x^{2 + 1} (| y | \sqrt[4]{x y^{2}})$

Step 12.3

Add $2$ and $1$ .

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

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