Simplify cube root of 1/(3x)

$\sqrt[3]{\frac{1}{3 x}}$

Step 1

Rewrite $\sqrt[3]{\frac{1}{3 x}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{3 x}}$ .

$\frac{\sqrt[3]{1}}{\sqrt[3]{3 x}}$

Step 2

Any root of $1$ is $1$ .

$\frac{1}{\sqrt[3]{3 x}}$

Step 3

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

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

Step 4

Combine and simplify the denominator.

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 5

Simplify the numerator.

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

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

Apply the product rule to $3 x$ .

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

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

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