Comparing big powers Which of the following is the largest?
A. $1^{200}$
B. $2^{400}$
C.$4^{80}$
D. $6^{300}$
E. $10^{250}$

I'm stuck trying to solve this. Obviously A and C are wrong ($4^{80}$ is less than $2^{400}$ and 1 to any power is always 1). And cancelling $2^{200}$ from each of the remaining choices, I can also eliminate B. However, I don't really know how to compare D and E... Any hints or helps?
$$6^{300} = (6^6)^{50} \ \ ; \ \ 10^{250} = (10^5)^{50}$$
so it is enough to check what is larger between $6^6$ and $10^5$. Now,
$$6^6 = (6^3)^2 = 216^2 &lt; 300^2 = 90000 &lt; 100000 = 10^5$$
$$2^{400} = 1024^{40} &lt; 10000^{40} = 10^{160} &lt; 10^{250}$$

See the other answers for $6^{300}$ and $10^{250}$.