Simplify ( fourth root of 32x^16)/( cube root of 27y^4)

$\frac{\sqrt[4]{32 x^{16}}}{\sqrt[3]{27 y^{4}}}$

Step 1

Simplify the numerator.

Rewrite $32 x^{16}$ as ${(2 x^{4})}^{4} \cdot 2$ .

Factor $16$ out of $32$ .

$\frac{\sqrt[4]{16 (2) x^{16}}}{\sqrt[3]{27 y^{4}}}$

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

$\frac{\sqrt[4]{2^{4} \cdot 2 x^{16}}}{\sqrt[3]{27 y^{4}}}$

Rewrite $x^{16}$ as ${(x^{4})}^{4}$ .

$\frac{\sqrt[4]{2^{4} \cdot 2 {(x^{4})}^{4}}}{\sqrt[3]{27 y^{4}}}$

Move $2$ .

$\frac{\sqrt[4]{2^{4} {(x^{4})}^{4} \cdot 2}}{\sqrt[3]{27 y^{4}}}$

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

$\frac{\sqrt[4]{{(2 x^{4})}^{4} \cdot 2}}{\sqrt[3]{27 y^{4}}}$

Pull terms out from under the radical.

$\frac{2 x^{4} \sqrt[4]{2}}{\sqrt[3]{27 y^{4}}}$

Step 2

Simplify the denominator.

Rewrite $27 y^{4}$ as ${(3 y)}^{3} y$ .

Rewrite $27$ as $3^{3}$ .

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

Factor out $y^{3}$ .

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

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

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

Pull terms out from under the radical.

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

Step 3

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

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

Step 4

Combine and simplify the denominator.

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

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

Move $\sqrt[3]{y}$ .

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

Combine $\frac{1}{3}$ and $3$ .

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 5

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

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

Multiply $y$ by $y$ .

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

Step 6

Simplify the numerator.

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

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

Combine exponents.

Rewrite the expression using the least common index of $12$ .

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

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

Rewrite $y^{\frac{2}{3}}$ as $y^{\frac{8}{12}}$ .

$\frac{2 x^{4} (y^{\frac{8}{12}} \sqrt[4]{2})}{3 y^{2}}$

Rewrite $y^{\frac{8}{12}}$ as $\sqrt[12]{y^{8}}$ .

$\frac{2 x^{4} (\sqrt[12]{y^{8}} \sqrt[4]{2})}{3 y^{2}}$

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

$\frac{2 x^{4} (\sqrt[12]{y^{8}} \cdot 2^{\frac{1}{4}})}{3 y^{2}}$

Rewrite $2^{\frac{1}{4}}$ as $2^{\frac{3}{12}}$ .

$\frac{2 x^{4} (\sqrt[12]{y^{8}} \cdot 2^{\frac{3}{12}})}{3 y^{2}}$

Rewrite $2^{\frac{3}{12}}$ as $\sqrt[12]{2^{3}}$ .

$\frac{2 x^{4} (\sqrt[12]{y^{8}} \sqrt[12]{2^{3}})}{3 y^{2}}$

Combine using the product rule for radicals.

$\frac{2 x^{4} \sqrt[12]{y^{8} \cdot 2^{3}}}{3 y^{2}}$

Raise $2$ to the power of $3$ .

$\frac{2 x^{4} \sqrt[12]{y^{8} \cdot 8}}{3 y^{2}}$

Step 7

Move $8$ to the left of $y^{8}$ .

$\frac{2 x^{4} \sqrt[12]{8 y^{8}}}{3 y^{2}}$

Simplify ( fourth root of 32x^16)/( cube root of 27y^4)

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