Simplify cube root of (2x)/(81y^12)

$\sqrt[3]{\frac{2 x}{81 y^{12}}}$

Step 1

Rewrite $\frac{2 x}{81 y^{12}}$ as ${(\frac{1}{3 y^{4}})}^{3} \frac{2 x}{3}$ .

Factor the perfect power $1^{3}$ out of $2 x$ .

$\sqrt[3]{\frac{1^{3} (2 x)}{81 y^{12}}}$

Factor the perfect power ${(3 y^{4})}^{3}$ out of $81 y^{12}$ .

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

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

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

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

Step 4

Combine.

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

Step 5

Multiply $\sqrt[3]{2 x}$ by $1$ .

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

Step 6

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

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

Step 7

Combine and simplify the denominator.

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

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

Move $\sqrt[3]{3}$ .

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 8

Simplify the numerator.

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

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

Raise $3$ to the power of $2$ .

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

Step 9

Multiply $3$ by $3$ .

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

Step 10

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $9$ by $2$ .

$\frac{\sqrt[3]{18 x}}{9 y^{4}}$

Simplify cube root of (2x)/(81y^12)

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