The volume of one cube is calculated by multiplying the length of one side by itself twice: $25^3=15625$. If Sam has 16 cubes, the total volume would be $15625 \cdot 16 = 250,000$. This can be written as a power of 2 by observing that $2^4= 2 \cdot 2 \cdot 2 \cdot 2 = 16$, so $250,000 = 16 \cdot 15625 = 2^4 \cdot 5^4$. Thus, the total volume can be written as $2^4 \cdot 5^4$. Answer: $\boxed{2^4 \cdot 5^4}$.
