Simplify fourth root of 4* cube root of 2

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

Rewrite $4$ as $2^{2}$ .

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

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

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

Pull terms out from under the radical, assuming positive real numbers.

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

Multiply $\sqrt{2} \sqrt[3]{2}$ .

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine using the product rule for radicals.

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

Multiply $2^{3}$ by $2^{2}$ by adding the exponents.

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

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

Add $3$ and $2$ .

$\sqrt[6]{2^{5}}$

Raise $2$ to the power of $5$ .

$\sqrt[6]{32}$

The result can be shown in multiple forms.

Exact Form:

$\sqrt[6]{32}$

Decimal Form:

$1.78179743 \dots$