Simplify ( square root of cube root of 64)^4

${(\sqrt{\sqrt[3]{64}})}^{4}$

Rewrite $64$ as $4^{3}$ .

${\sqrt{\sqrt[3]{4^{3}}}}^{4}$

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

${\sqrt{4}}^{4}$

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

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

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

$2^{4}$

Raise $2$ to the power of $4$ .

$16$