TITLE: Put $2^{600}$, $3^{500}$, $4^{400}$, $5^{300}$, and $6^{200}$ in order from least to greatest
QUESTION [7 upvotes]: Put $2^{600}$, $3^{500}$, $4^{400}$, $5^{300}$, and $6^{200}$ in order.

Problem I found while looking at old problems from math competitions. 
Clearly a simple solution would be to compare $600\ln2$, $500\ln3$, etc.
But how would one go about solving this problem without a calculator? Expressing the numbers in the same base somehow?
Would appreciate any insight.

REPLY [2 votes]: To evaluate these values, they must be placed on the same “platform”. 
Re-writing $2^{600}$ as $(2^6)^{100}$; and $3^{500}$ as $(3^5)^{100}$ and so on will make them comparable (now), under the same platform.
After doing that, then, you are comparing $(64)^{100}, (243)^{100}$ and so on.