$$
2^{\sqrt{200}} = 2^{\sqrt{100}*\sqrt{2}} = 2^{10*\sqrt{2}} = \sqrt{2}^{2*10*\sqrt{2}} = \sqrt{2}^{5*4*\sqrt{2}} = (4*\sqrt{2})^{4*\sqrt{2}} = a^a
$$

Hence, $$a=4*\sqrt{2}$$ and $$a^2=32$$.

Scroogey

Solving this problem seems to be a door to a Wall St Job (

science

If
$$
a^a = 2^\sqrt{200}
$$
then
$$
a^2 =\ ?
$$

Previous iteration: https://stacker.news/items/823793/r/south_korea_ln (answer in https://stacker.news/items/823793/r/south_korea_ln?commentId=823807). 
Solving this problem seems to be a door to a Wall St Job (https://stacker.news/items/823793/r/south_korea_ln?commentId=823870)~~