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

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

Step 1

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

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

Step 2

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

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

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

Factor $y$ out of $x y$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Rewrite $\sqrt[3]{\frac{x}{4 y}}$ as $\frac{\sqrt[3]{x}}{\sqrt[3]{4 y}}$ .

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

Step 5

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

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

Step 6

Combine and simplify the denominator.

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

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

Raise $\sqrt[3]{4 y}$ to the power of $1$ .

$2 \frac{\sqrt[3]{x} {\sqrt[3]{4 y}}^{2}}{{\sqrt[3]{4 y}}^{1} {\sqrt[3]{4 y}}^{2}}$

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

$2 \frac{\sqrt[3]{x} {\sqrt[3]{4 y}}^{2}}{{\sqrt[3]{4 y}}^{1 + 2}}$

Add $1$ and $2$ .

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

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

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

$2 \frac{\sqrt[3]{x} {\sqrt[3]{4 y}}^{2}}{{({(4 y)}^{\frac{1}{3}})}^{3}}$

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

$2 \frac{\sqrt[3]{x} {\sqrt[3]{4 y}}^{2}}{{(4 y)}^{1}}$

Simplify.

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

Step 7

Simplify the numerator.

Rewrite ${\sqrt[3]{4 y}}^{2}$ as $\sqrt[3]{{(4 y)}^{2}}$ .

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

Apply the product rule to $4 y$ .

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

Raise $4$ to the power of $2$ .

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

Rewrite $16 y^{2}$ as $2^{3} \cdot (2 y^{2})$ .

Factor $8$ out of $16$ .

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

Rewrite $8$ as $2^{3}$ .

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

Add parentheses.

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

Pull terms out from under the radical.

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

Combine using the product rule for radicals.

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

Step 8

Reduce the expression by cancelling the common factors.

Cancel the common factor of $2$ .

Factor $2$ out of $4 y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Reorder factors in $\frac{\sqrt[3]{x \cdot 2 y^{2}}}{y}$ .

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

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

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