Simplify fifth root of 4* square root of 2

$\sqrt[5]{4} \cdot \sqrt{2}$

Multiply $\sqrt[5]{4} \sqrt{2}$ .

Rewrite the expression using the least common index of $10$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{4}$ as $4^{\frac{1}{5}}$ .

$4^{\frac{1}{5}} \sqrt{2}$

Rewrite $4^{\frac{1}{5}}$ as $4^{\frac{2}{10}}$ .

$4^{\frac{2}{10}} \sqrt{2}$

Rewrite $4^{\frac{2}{10}}$ as $\sqrt[10]{4^{2}}$ .

$\sqrt[10]{4^{2}} \sqrt{2}$

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

$\sqrt[10]{4^{2}} \cdot 2^{\frac{1}{2}}$

Rewrite $2^{\frac{1}{2}}$ as $2^{\frac{5}{10}}$ .

$\sqrt[10]{4^{2}} \cdot 2^{\frac{5}{10}}$

Rewrite $2^{\frac{5}{10}}$ as $\sqrt[10]{2^{5}}$ .

$\sqrt[10]{4^{2}} \sqrt[10]{2^{5}}$

Combine using the product rule for radicals.

$\sqrt[10]{4^{2} \cdot 2^{5}}$

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

$\sqrt[10]{{(2^{2})}^{2} \cdot 2^{5}}$

Multiply the exponents in ${(2^{2})}^{2}$ .

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

$\sqrt[10]{2^{2 \cdot 2} \cdot 2^{5}}$

Multiply $2$ by $2$ .

$\sqrt[10]{2^{4} \cdot 2^{5}}$

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

$\sqrt[10]{2^{4 + 5}}$

Add $4$ and $5$ .

$\sqrt[10]{2^{9}}$

Raise $2$ to the power of $9$ .

$\sqrt[10]{512}$

The result can be shown in multiple forms.

Exact Form:

$\sqrt[10]{512}$

Decimal Form:

$1.86606598 \dots$