What is the largest number among the following?

Correct Answer: (A) 2^40

Expressing the numbers in base 2 powers where possible: $4^{18} = (2^2)^{18} = 2^{36}$ and $8^{12} = (2^3)^{12} = 2^{36}$, both of which are strictly smaller than $2^{40}$. Comparing $2^{40}$ with $3^{21}$: $2^{40} = (2^4)^{10} = 16^{10}$, whereas $3^{21} = 3$$\times$ (3^2)^{10} = 3 $\times$ 9^{10}. Since $16^{10} = (1.777...)^{10}$$\times$ 9^{10} $\approx$ 317 $\times$ 9^{10} > 3 $\times$ 9^{10}, $2^{40}$ is strictly the largest.