It's quite easy to observe that for $k \ge 0$:

$$
\begin{align}
2^{2^k} &= 4, 16, 256, 65536, \dots\\
\sqrt{2^{2^k}} &= 2, 4, 16, 256,\dots
\end{align}
$$

More in general:

$$
\sqrt{2^{2^k}} = 2^{2^{k-1}}
$$

How can I prove this identity?