Simplify fourth root of 2* eighth root of 4

$\sqrt[4]{2} \cdot \sqrt[8]{4}$

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

$\sqrt[4]{2} \cdot \sqrt[8]{2^{2}}$

Rewrite $\sqrt[8]{2^{2}}$ as $\sqrt[4]{\sqrt{2^{2}}}$ .

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

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

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

Combine using the product rule for radicals.

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

Write the expression using exponents.

Multiply $2$ by $2$ .

$\sqrt[4]{4}$

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

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

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

$\sqrt{\sqrt{2^{2}}}$

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

$\sqrt{2}$

The result can be shown in multiple forms.

Exact Form:

$\sqrt{2}$

Decimal Form:

$1.41421356 \dots$