Simplify the Radical Expression cube root of (x^2)/(9y)

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

Step 1

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

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 4

Simplify the numerator.

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

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

Apply the product rule to $9 y$ .

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

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

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

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

Factor $27$ out of $81$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Combine using the product rule for radicals.

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

Step 5

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

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

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

Cancel the common factors.

Factor $3$ out of $9 y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

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

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

Simplify the Radical Expression cube root of (x^2)/(9y)

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