Simplify the Radical Expression fourth root of (32x^6)/(y^2)

$\sqrt[4]{\frac{32 x^{6}}{y^{2}}}$

Rewrite $\sqrt[4]{\frac{32 x^{6}}{y^{2}}}$ as $\frac{\sqrt[4]{32 x^{6}}}{\sqrt[4]{y^{2}}}$ .

$\frac{\sqrt[4]{32 x^{6}}}{\sqrt[4]{y^{2}}}$

Simplify the numerator.

Rewrite $32 x^{6}$ as ${(2 x)}^{4} \cdot (2 x^{2})$ .

Factor $16$ out of $32$ .

$\frac{\sqrt[4]{16 (2) x^{6}}}{\sqrt[4]{y^{2}}}$

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

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

Factor out $x^{4}$ .

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

Move $2$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Remove non-negative terms from the absolute value.

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

Simplify the denominator.

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

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

Pull terms out from under the radical.

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

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

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

Combine and simplify the denominator.

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

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

Raise $\sqrt{| y |}$ to the power of $1$ .

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

Raise $\sqrt{| y |}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{| y |}}^{2}$ as $| y |$ .

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

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

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

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

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the numerator.

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

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

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

Rewrite ${| y |}^{\frac{1}{2}}$ as ${| y |}^{\frac{2}{4}}$ .

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

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

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

Combine using the product rule for radicals.

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

Remove the absolute value in ${| y |}^{2}$ because exponentiations with even powers are always positive.

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

Move $2$ to the left of $y^{2}$ .

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

Simplify the Radical Expression fourth root of (32x^6)/(y^2)

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