Simplify 4x^3 square root of 64x^5

$4 x^{3} \sqrt{64 x^{5}}$

Rewrite $64 x^{5}$ as ${(8 x^{2})}^{2} x$ .

Rewrite $64$ as $8^{2}$ .

$4 x^{3} \sqrt{8^{2} x^{5}}$

Factor out $x^{4}$ .

$4 x^{3} \sqrt{8^{2} (x^{4} x)}$

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

$4 x^{3} \sqrt{8^{2} ({(x^{2})}^{2} x)}$

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

$4 x^{3} \sqrt{{(8 x^{2})}^{2} x}$

Pull terms out from under the radical.

$4 x^{3} (8 x^{2} \sqrt{x})$

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

Move $x^{2}$ .

$4 (x^{2} x^{3}) (8 \sqrt{x})$

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

$4 x^{2 + 3} (8 \sqrt{x})$

Add $2$ and $3$ .

$4 x^{5} (8 \sqrt{x})$

Multiply $8$ by $4$ .

$32 x^{5} \sqrt{x}$