<s>[INST] $\sqrt[5]{2^4\sqrt[3]{16}} = ? $ $$\sqrt[5]{2^4\sqrt[3]{16}} = ? $$
Rewriting this and we have
$$\sqrt[5\cdot 3]{2^{4\cdot 3}4}$$
$$\sqrt[15]{2^{12}2^2}$$
Finally we get
$$\sqrt[15]{2^{12}2^2} = \sqrt[15]{2^{14}} = 2^{\frac{15}{14}}$$
Am I right?
 [/INST] In your first reformulation, you should get $\sqrt[15]{2^{4 \cdot 3} \cdot 16}$ and in the last part note that $\sqrt[a]{c^b} = c^{\frac{b}{a}}$ and not $c^{\frac{a}{b}}$.
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