Simplify ( cube root of 4xy^7)/( cube root of 32x^4y^4)

$\frac{\sqrt[3]{4 x y^{7}}}{\sqrt[3]{32 x^{4} y^{4}}}$

Step 1

Combine $\sqrt[3]{4 x y^{7}}$ and $\sqrt[3]{32 x^{4} y^{4}}$ into a single radical.

$\sqrt[3]{\frac{4 x y^{7}}{32 x^{4} y^{4}}}$

Step 2

Reduce the expression $\frac{4 x y^{7}}{32 x^{4} y^{4}}$ by cancelling the common factors.

Factor $4$ out of $4 x y^{7}$ .

$\sqrt[3]{\frac{4 (x y^{7})}{32 x^{4} y^{4}}}$

Factor $4$ out of $32 x^{4} y^{4}$ .

$\sqrt[3]{\frac{4 (x y^{7})}{4 (8 x^{4} y^{4})}}$

Cancel the common factor.

$\sqrt[3]{\frac{4 (x y^{7})}{4 (8 x^{4} y^{4})}}$

Rewrite the expression.

$\sqrt[3]{\frac{x y^{7}}{8 x^{4} y^{4}}}$

Step 3

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

Factor $x$ out of $x y^{7}$ .

$\sqrt[3]{\frac{x (y^{7})}{8 x^{4} y^{4}}}$

Cancel the common factors.

Factor $x$ out of $8 x^{4} y^{4}$ .

$\sqrt[3]{\frac{x (y^{7})}{x (8 x^{3} y^{4})}}$

Cancel the common factor.

$\sqrt[3]{\frac{x y^{7}}{x (8 x^{3} y^{4})}}$

Rewrite the expression.

$\sqrt[3]{\frac{y^{7}}{8 x^{3} y^{4}}}$

Step 4

Cancel the common factor of $y^{7}$ and $y^{4}$ .

Factor $y^{4}$ out of $y^{7}$ .

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

Cancel the common factors.

Factor $y^{4}$ out of $8 x^{3} y^{4}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

Rewrite $8 x^{3}$ as ${(2 x)}^{3}$ .

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

Step 6

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

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

Step 7

Pull terms out from under the radical, assuming real numbers.

$\frac{y}{2 x}$