Simplify cube root of 3* sixth root of 2

$\sqrt[3]{3} \cdot \sqrt[6]{2}$

Rewrite the expression using the least common index of $6$ .

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

$3^{\frac{1}{3}} \sqrt[6]{2}$

Rewrite $3^{\frac{1}{3}}$ as $3^{\frac{2}{6}}$ .

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

Rewrite $3^{\frac{2}{6}}$ as $\sqrt[6]{3^{2}}$ .

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

Combine using the product rule for radicals.

$\sqrt[6]{3^{2} \cdot 2}$

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

$\sqrt[6]{9 \cdot 2}$

Multiply $9$ by $2$ .

$\sqrt[6]{18}$

The result can be shown in multiple forms.

Exact Form:

$\sqrt[6]{18}$

Decimal Form:

$1.61887040 \dots$