Simplify the Radical Expression ( cube root of 3)/( cube root of 2x)

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

Step 1

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

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

Step 2

Combine and simplify the denominator.

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 3

Simplify the numerator.

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

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

Apply the product rule to $2 x$ .

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

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

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

Step 4

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $3$ by $4$ .

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