$sqrt {{2^?}} = left( {{8^2} imes {5^2}} ight) div left( {200sqrt 2 } ight)$

$sqrt {{2^?}} = frac{{{{left( {{2^3}} ight)}^2} imes {5^2}}}{{200 imes sqrt 2 }}$

$Rightarrow sqrt {{2^?}} = frac{{{2^6} imes 25}}{{200 imes sqrt 2 }}$

$Rightarrow {(2)^{frac{?}{2}}} = frac{{{2^6}}}{{8sqrt 2 }} = frac{{{2^3}}}{{sqrt 2 }} = {(2)^{frac{5}{2}}}$

$Rightarrow frac{?}{2} = frac{5}{2} Rightarrow ? = 5$