Simplify (xy)/( cube root of 9xy^2)

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

Step 1

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

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

Step 2

Simplify terms.

Combine and simplify the denominator.

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

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

Raise $\sqrt[3]{9 x y^{2}}$ to the power of $1$ .

$\frac{x y {\sqrt[3]{9 x y^{2}}}^{2}}{{\sqrt[3]{9 x y^{2}}}^{1} {\sqrt[3]{9 x y^{2}}}^{2}}$

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

$\frac{x y {\sqrt[3]{9 x y^{2}}}^{2}}{{\sqrt[3]{9 x y^{2}}}^{1 + 2}}$

Add $1$ and $2$ .

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

Rewrite ${\sqrt[3]{9 x y^{2}}}^{3}$ as $9 x y^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{9 x y^{2}}$ as ${(9 x y^{2})}^{\frac{1}{3}}$ .

$\frac{x y {\sqrt[3]{9 x y^{2}}}^{2}}{{({(9 x y^{2})}^{\frac{1}{3}})}^{3}}$

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

$\frac{x y {\sqrt[3]{9 x y^{2}}}^{2}}{{(9 x y^{2})}^{\frac{1}{3} \cdot 3}}$

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{x y {\sqrt[3]{9 x y^{2}}}^{2}}{{(9 x y^{2})}^{1}}$

Simplify.

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

Cancel the common factor of $x$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

Factor $y$ out of $y {\sqrt[3]{9 x y^{2}}}^{2}$ .

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

Cancel the common factors.

Factor $y$ out of $9 y^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

Simplify the numerator.

Rewrite ${\sqrt[3]{9 x y^{2}}}^{2}$ as ${({(9 x y^{2})}^{2})}^{\frac{1}{3}}$ .

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

Apply the product rule to $9 x y^{2}$ .

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

Apply the product rule to $9 x$ .

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

Raise $9$ to the power of $2$ .

$\frac{\sqrt[3]{81 x^{2} {(y^{2})}^{2}}}{9 y}$

Multiply the exponents in ${(y^{2})}^{2}$ .

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

$\frac{\sqrt[3]{81 x^{2} y^{2 \cdot 2}}}{9 y}$

Multiply $2$ by $2$ .

$\frac{\sqrt[3]{81 x^{2} y^{4}}}{9 y}$

Rewrite $81 x^{2} y^{4}$ as ${(3 y)}^{3} \cdot (3 x^{2} y)$ .

Factor $27$ out of $81$ .

$\frac{\sqrt[3]{27 (3) x^{2} y^{4}}}{9 y}$

Rewrite $27$ as $3^{3}$ .

$\frac{\sqrt[3]{3^{3} \cdot 3 x^{2} y^{4}}}{9 y}$

Factor out $y^{3}$ .

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

Move $x^{2}$ .

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

Move $3$ .

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

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

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

Add parentheses.

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

Add parentheses.

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

Pull terms out from under the radical.

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

Step 4

Reduce the expression by cancelling the common factors.

Cancel the common factor of $3$ and $9$ .

Factor $3$ out of $3 y \sqrt[3]{3 x^{2} y}$ .

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

Cancel the common factors.

Factor $3$ out of $9 y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $y$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (xy)/( cube root of 9xy^2)

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