Simplify square root of 4x^11* fourth root of 16x^8

$\sqrt{4 x^{11}} \cdot \sqrt[4]{16 x^{8}}$

Rewrite $4 x^{11}$ as ${(2 x^{5})}^{2} x$ .

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

$\sqrt{2^{2} x^{11}} \cdot \sqrt[4]{16 x^{8}}$

Factor out $x^{10}$ .

$\sqrt{2^{2} (x^{10} x)} \cdot \sqrt[4]{16 x^{8}}$

Rewrite $x^{10}$ as ${(x^{5})}^{2}$ .

$\sqrt{2^{2} ({(x^{5})}^{2} x)} \cdot \sqrt[4]{16 x^{8}}$

Rewrite $2^{2} {(x^{5})}^{2}$ as ${(2 x^{5})}^{2}$ .

$\sqrt{{(2 x^{5})}^{2} x} \cdot \sqrt[4]{16 x^{8}}$

Pull terms out from under the radical.

$2 x^{5} \sqrt{x} \cdot \sqrt[4]{16 x^{8}}$

Rewrite $16 x^{8}$ as ${(2 x^{2})}^{4}$ .

$2 x^{5} \sqrt{x} \cdot \sqrt[4]{{(2 x^{2})}^{4}}$

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

$2 x^{5} \sqrt{x} \cdot (2 x^{2})$

Multiply $x^{5}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$2 (x^{2} x^{5}) \sqrt{x} \cdot 2$

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

$2 x^{2 + 5} \sqrt{x} \cdot 2$

Add $2$ and $5$ .

$2 x^{7} \sqrt{x} \cdot 2$

Multiply $2$ by $2$ .

$4 x^{7} \sqrt{x}$