Simplify ( cube root of 24x^2y^5)/( cube root of 3x^5y^2)

$\frac{\sqrt[3]{24 x^{2} y^{5}}}{\sqrt[3]{3 x^{5} y^{2}}}$

Step 1

Combine $\sqrt[3]{24 x^{2} y^{5}}$ and $\sqrt[3]{3 x^{5} y^{2}}$ into a single radical.

$\sqrt[3]{\frac{24 x^{2} y^{5}}{3 x^{5} y^{2}}}$

Step 2

Reduce the expression $\frac{24 x^{2} y^{5}}{3 x^{5} y^{2}}$ by cancelling the common factors.

Factor $3$ out of $24 x^{2} y^{5}$ .

$\sqrt[3]{\frac{3 (8 x^{2} y^{5})}{3 x^{5} y^{2}}}$

Factor $3$ out of $3 x^{5} y^{2}$ .

$\sqrt[3]{\frac{3 (8 x^{2} y^{5})}{3 (x^{5} y^{2})}}$

Cancel the common factor.

$\sqrt[3]{\frac{3 (8 x^{2} y^{5})}{3 (x^{5} y^{2})}}$

Rewrite the expression.

$\sqrt[3]{\frac{8 x^{2} y^{5}}{x^{5} y^{2}}}$

Step 3

Cancel the common factor of $x^{2}$ and $x^{5}$ .

Factor $x^{2}$ out of $8 x^{2} y^{5}$ .

$\sqrt[3]{\frac{x^{2} (8 y^{5})}{x^{5} y^{2}}}$

Cancel the common factors.

Factor $x^{2}$ out of $x^{5} y^{2}$ .

$\sqrt[3]{\frac{x^{2} (8 y^{5})}{x^{2} (x^{3} y^{2})}}$

Cancel the common factor.

$\sqrt[3]{\frac{x^{2} (8 y^{5})}{x^{2} (x^{3} y^{2})}}$

Rewrite the expression.

$\sqrt[3]{\frac{8 y^{5}}{x^{3} y^{2}}}$

Step 4

Cancel the common factor of $y^{5}$ and $y^{2}$ .

Factor $y^{2}$ out of $8 y^{5}$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

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

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

Step 6

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

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

Step 7

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

$\frac{2 y}{x}$