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

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

Combine $\sqrt[3]{2 x y^{3}}$ and $\sqrt[3]{4 x^{2} y}$ into a single radical.

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

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

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

Cancel the common factors.

Factor $y$ out of $2 x y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

Combine and simplify the denominator.

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

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

Raise $\sqrt[3]{2 x}$ to the power of $1$ .

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

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

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

Add $1$ and $2$ .

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

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2 x}$ as ${(2 x)}^{\frac{1}{3}}$ .

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the numerator.

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

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

Apply the product rule to $2 x$ .

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

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

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

Combine using the product rule for radicals.

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{\sqrt[3]{y^{2} \cdot 4 x^{2}}}{2 x}$ .

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

Apply the product rule to $2 x$ .

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

Simplify the numerator.

Rewrite ${\sqrt[3]{y^{2} \cdot 4 x^{2}}}^{3}$ as $y^{2} \cdot 4 x^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y^{2} \cdot 4 x^{2}}$ as ${(y^{2} \cdot 4 x^{2})}^{\frac{1}{3}}$ .

$\frac{{({(y^{2} \cdot 4 x^{2})}^{\frac{1}{3}})}^{3}}{2^{3} x^{3}}$

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

$\frac{{(y^{2} \cdot 4 x^{2})}^{\frac{1}{3} \cdot 3}}{2^{3} x^{3}}$

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

$\frac{{(y^{2} \cdot 4 x^{2})}^{\frac{3}{3}}}{2^{3} x^{3}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{{(y^{2} \cdot 4 x^{2})}^{\frac{3}{3}}}{2^{3} x^{3}}$

Rewrite the expression.

$\frac{{(y^{2} \cdot 4 x^{2})}^{1}}{2^{3} x^{3}}$

Simplify.

$\frac{y^{2} \cdot 4 x^{2}}{2^{3} x^{3}}$

Move $4$ to the left of $y^{2}$ .

$\frac{4 y^{2} x^{2}}{2^{3} x^{3}}$

Reduce the expression by cancelling the common factors.

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

$\frac{4 y^{2} x^{2}}{8 x^{3}}$

Cancel the common factor of $4$ and $8$ .

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

$\frac{4 (y^{2} x^{2})}{8 x^{3}}$

Cancel the common factors.

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

$\frac{4 (y^{2} x^{2})}{4 (2 x^{3})}$

Cancel the common factor.

$\frac{4 (y^{2} x^{2})}{4 (2 x^{3})}$

Rewrite the expression.

$\frac{y^{2} x^{2}}{2 x^{3}}$

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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

$\frac{x^{2} y^{2}}{2 x^{3}}$

Cancel the common factors.

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

$\frac{x^{2} y^{2}}{x^{2} (2 x)}$

Cancel the common factor.

$\frac{x^{2} y^{2}}{x^{2} (2 x)}$

Rewrite the expression.

$\frac{y^{2}}{2 x}$

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

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