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

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

Step 1

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

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 4

Simplify the numerator.

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

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

Apply the product rule to $2 a$ .

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

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

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

Step 5

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $3$ by $4$ .

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