Simplify cube root of x/(y^2)

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

Step 1

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

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

Combine $\frac{2}{3}$ and $3$ .

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

Multiply $2$ by $3$ .

$\frac{\sqrt[3]{x} {\sqrt[3]{y^{2}}}^{2}}{y^{\frac{6}{3}}}$

Cancel the common factor of $6$ and $3$ .

Factor $3$ out of $6$ .

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

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2$ by $1$ .

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

Step 4

Simplify the numerator.

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

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

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

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

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

Multiply $2$ by $2$ .

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

Factor out $y^{3}$ .

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

Pull terms out from under the radical.

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

Combine using the product rule for radicals.

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

Step 5

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify cube root of x/(y^2)

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