Simplify the Radical Expression (x^4y^2)/( fourth root of x^3y^2)

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

Add $1$ and $3$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $4$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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Cancel the common factor.

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

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

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

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

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

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

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

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

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

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

Multiply $3$ by $3$ .

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

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

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

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

Multiply $2$ by $3$ .

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

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

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Factor out $x^{8}$ .

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

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

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

Factor out $y^{4}$ .

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

Move $x$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Remove non-negative terms from the absolute value.

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

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

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Move $x^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

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

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