Simplify cube root of (2x)/(3y)

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

Rewrite $\sqrt[3]{\frac{2 x}{3 y}}$ as $\frac{\sqrt[3]{2 x}}{\sqrt[3]{3 y}}$ .

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the numerator.

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

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

Apply the product rule to $3 y$ .

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

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

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

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $9$ by $2$ .

$\frac{\sqrt[3]{18 x y^{2}}}{3 y}$