2010/11 AHSMC Part 1 Problem 11

The first three choices are equal respectively to $2^{2^{256}}$, $2^{2^{512}}$ and $2^{2^{81}}$. Clearly, the second one is the largest among them. The fourth number is equal to $2^{3^{16}}$. Since $2^{512} = 4^{256} > 3^{256} > 3^{16}$, the second number is larger than the fourth one. The fifth number is equal to $3^{2^{16}}$. Clearly, $2^{2^{256}} = 4^{2^{255}} > 3^{2^{255}} > 3^{2^{16}}$. Hence the second number is the largest among the five choices. The answer is (B).