Simplify cube root of square root of 64x^3

$\sqrt[3]{\sqrt{64 x^{3}}}$

Rewrite $64 x^{3}$ as ${(8 x)}^{2} x$ .

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

$\sqrt[3]{\sqrt{8^{2} x^{3}}}$

Factor out $x^{2}$ .

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

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

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

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

Pull terms out from under the radical.

$\sqrt[3]{8 x \sqrt{x}}$

Rewrite $8 x \sqrt{x}$ as $2^{3} (x \sqrt{x})$ .

Rewrite $8$ as $2^{3}$ .

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

Add parentheses.

$\sqrt[3]{2^{3} (x \sqrt{x})}$

$\sqrt[3]{2^{3} (x \sqrt{x})}$

Pull terms out from under the radical.

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

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