Simplify ( cube root of 32x^3y^6)/( cube root of 2x^9y^2)

$\frac{\sqrt[3]{32 x^{3} y^{6}}}{\sqrt[3]{2 x^{9} y^{2}}}$

Step 1

Combine $\sqrt[3]{32 x^{3} y^{6}}$ and $\sqrt[3]{2 x^{9} y^{2}}$ into a single radical.

$\sqrt[3]{\frac{32 x^{3} y^{6}}{2 x^{9} y^{2}}}$

Step 2

Reduce the expression $\frac{32 x^{3} y^{6}}{2 x^{9} y^{2}}$ by cancelling the common factors.

Factor $2$ out of $32 x^{3} y^{6}$ .

$\sqrt[3]{\frac{2 (16 x^{3} y^{6})}{2 x^{9} y^{2}}}$

Factor $2$ out of $2 x^{9} y^{2}$ .

$\sqrt[3]{\frac{2 (16 x^{3} y^{6})}{2 (x^{9} y^{2})}}$

Cancel the common factor.

$\sqrt[3]{\frac{2 (16 x^{3} y^{6})}{2 (x^{9} y^{2})}}$

Rewrite the expression.

$\sqrt[3]{\frac{16 x^{3} y^{6}}{x^{9} y^{2}}}$

Step 3

Cancel the common factor of $x^{3}$ and $x^{9}$ .

Factor $x^{3}$ out of $16 x^{3} y^{6}$ .

$\sqrt[3]{\frac{x^{3} (16 y^{6})}{x^{9} y^{2}}}$

Cancel the common factors.

Factor $x^{3}$ out of $x^{9} y^{2}$ .

$\sqrt[3]{\frac{x^{3} (16 y^{6})}{x^{3} (x^{6} y^{2})}}$

Cancel the common factor.

$\sqrt[3]{\frac{x^{3} (16 y^{6})}{x^{3} (x^{6} y^{2})}}$

Rewrite the expression.

$\sqrt[3]{\frac{16 y^{6}}{x^{6} y^{2}}}$

Step 4

Cancel the common factor of $y^{6}$ and $y^{2}$ .

Factor $y^{2}$ out of $16 y^{6}$ .

$\sqrt[3]{\frac{y^{2} (16 y^{4})}{x^{6} y^{2}}}$

Cancel the common factors.

Factor $y^{2}$ out of $x^{6} y^{2}$ .

$\sqrt[3]{\frac{y^{2} (16 y^{4})}{y^{2} x^{6}}}$

Cancel the common factor.

$\sqrt[3]{\frac{y^{2} (16 y^{4})}{y^{2} x^{6}}}$

Rewrite the expression.

$\sqrt[3]{\frac{16 y^{4}}{x^{6}}}$

Step 5

Rewrite $\frac{16 y^{4}}{x^{6}}$ as ${(\frac{2 y}{x^{2}})}^{3} (2 y)$ .

Factor the perfect power ${(2 y)}^{3}$ out of $16 y^{4}$ .

$\sqrt[3]{\frac{{(2 y)}^{3} \cdot (2 y)}{x^{6}}}$

Factor the perfect power ${(x^{2})}^{3}$ out of $x^{6}$ .

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

Rearrange the fraction $\frac{{(2 y)}^{3} \cdot (2 y)}{{(x^{2})}^{3} \cdot 1}$ .

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

Step 6

Pull terms out from under the radical.

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

Step 7

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

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

Simplify ( cube root of 32x^3y^6)/( cube root of 2x^9y^2)

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