smc

We can rewrite this as $2^{32}{5}^{25}$. We can also combine some of the factors to introduce factors of $10$, whose digit count is simple to evaluate because it simply adds $0$s. Thus, we have $2^{32}{5}^{25}=2^7{2}^{25}{5}^{25}=2^{7}10^{25}$. We can see that this final number is $2^7$ with $25$ $0$s annexed onto it. $2^7=128$, which has $3$ digits, so the entire number has $25+3=28$ digits, $\boxed{28}$.