Simplify cube root of 1/(108y^2)

$\sqrt[3]{\frac{1}{108 y^{2}}}$

Rewrite $\frac{1}{108 y^{2}}$ as ${(\frac{1}{3})}^{3} \frac{1}{4 y^{2}}$ .

Factor the perfect power $1^{3}$ out of $1$ .

$\sqrt[3]{\frac{1^{3} \cdot 1}{108 y^{2}}}$

Factor the perfect power $3^{3}$ out of $108 y^{2}$ .

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

Rearrange the fraction $\frac{1^{3} \cdot 1}{3^{3} \cdot (4 y^{2})}$ .

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

Pull terms out from under the radical.

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

Rewrite $\sqrt[3]{\frac{1}{4 y^{2}}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{4 y^{2}}}$ .

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

Combine.

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

Multiply $\sqrt[3]{1}$ by $1$ .

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

Any root of $1$ is $1$ .

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

Multiply $\frac{1}{3 \sqrt[3]{4 y^{2}}}$ by $\frac{{\sqrt[3]{4 y^{2}}}^{2}}{{\sqrt[3]{4 y^{2}}}^{2}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{1}{3 \sqrt[3]{4 y^{2}}}$ by $\frac{{\sqrt[3]{4 y^{2}}}^{2}}{{\sqrt[3]{4 y^{2}}}^{2}}$ .

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

Move $\sqrt[3]{4 y^{2}}$ .

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

Raise $\sqrt[3]{4 y^{2}}$ to the power of $1$ .

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the numerator.

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

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

Apply the product rule to $4 y^{2}$ .

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

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

$\frac{\sqrt[3]{16 {(y^{2})}^{2}}}{3 \cdot 4 y^{2}}$

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

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

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

Multiply $2$ by $2$ .

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

Rewrite $16 y^{4}$ as ${(2 y)}^{3} \cdot (2 y)$ .

Factor $8$ out of $16$ .

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

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

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

Factor out $y^{3}$ .

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

Move $2$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Reduce the expression by cancelling the common factors.

Multiply $3$ by $4$ .

$\frac{2 y \sqrt[3]{2 y}}{12 y^{2}}$

Cancel the common factor of $2$ and $12$ .

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

$\frac{2 (y \sqrt[3]{2 y})}{12 y^{2}}$

Cancel the common factors.

Factor $2$ out of $12 y^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

Cancel the common factors.

Factor $y$ out of $6 y^{2}$ .

$\frac{y (\sqrt[3]{2 y})}{y (6 y)}$

Cancel the common factor.

$\frac{y \sqrt[3]{2 y}}{y (6 y)}$

Rewrite the expression.

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

Simplify cube root of 1/(108y^2)

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