imc

Taking the root, we get $N=\sqrt{25^{64}\cdot 64^{25}}=5^{64}\cdot 8^{25}$. Now, we have $N=5^{64}\cdot 8^{25}=5^{64}\cdot (2^3{)}^{25}=5^{64}\cdot 2^{75}$. Combining the $2$'s and $5$'s gives us $(2\cdot 5{)}^{64}\cdot 2^{(75-64)}=(2\cdot 5{)}^{64}\cdot 2^{11}=10^{64}\cdot 2^{11}$. This is the number $2048$ with a string of sixty-four $0$'s at the end. Thus, the sum of the digits of $N$ is $2+4+8=14⟹\boxed{14}$