Simplify sixth root of 3* cube root of 4

$\sqrt[6]{3} \cdot \sqrt[3]{4}$

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$\sqrt[6]{3} \cdot 4^{\frac{1}{3}}$

Rewrite $4^{\frac{1}{3}}$ as $4^{\frac{2}{6}}$ .

$\sqrt[6]{3} \cdot 4^{\frac{2}{6}}$

Rewrite $4^{\frac{2}{6}}$ as $\sqrt[6]{4^{2}}$ .

$\sqrt[6]{3} \sqrt[6]{4^{2}}$

Combine using the product rule for radicals.

$\sqrt[6]{3 \cdot 4^{2}}$

Raise $4$ to the power of $2$ .

$\sqrt[6]{3 \cdot 16}$

Multiply $3$ by $16$ .

$\sqrt[6]{48}$

The result can be shown in multiple forms.

Exact Form:

$\sqrt[6]{48}$

Decimal Form:

$1.90636858 \dots$