Simplify cube root of (3xy^2)/(81x^4y^2)

$\sqrt[3]{\frac{3 x y^{2}}{81 x^{4} y^{2}}}$

Step 1

Reduce the expression $\frac{3 x y^{2}}{81 x^{4} y^{2}}$ by cancelling the common factors.

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

$\sqrt[3]{\frac{3 (x y^{2})}{81 x^{4} y^{2}}}$

Factor $3$ out of $81 x^{4} y^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 2

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

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

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

Cancel the common factors.

Factor $x$ out of $27 x^{4} y^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

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

Cancel the common factor.

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

Rewrite the expression.

$\sqrt[3]{\frac{1}{27 x^{3}}}$

Step 4

Rewrite $1$ as $1^{3}$ .

$\sqrt[3]{\frac{1^{3}}{27 x^{3}}}$

Step 5

Rewrite $27 x^{3}$ as ${(3 x)}^{3}$ .

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

Step 6

Rewrite $\frac{1^{3}}{{(3 x)}^{3}}$ as ${(\frac{1}{3 x})}^{3}$ .

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

Step 7

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

$\frac{1}{3 x}$