TITLE: Which is greater, $300 !$ or $(300^{300})^\frac {1}{2}$?
QUESTION [12 upvotes]: Which is greater among $300 !$ and $\sqrt {300^{300}}$ ?

The answer is $300 !$ (my textbook's answer). I do not know how to solve problems involving such large numbers.

REPLY [4 votes]: When calculations involve large factorials, the use of Stirling approximation for $n!$ is very convenient. In your case, you compare $n!$ to $n^{n/2}$. We can take logarithms of both sides and use the fact that Stirling approximation gives $\log(n!)$ close to $n \log(n) - n$; the logarithm of the second term is simply $\dfrac{n \log(n)} {2}$. Using this approximation, you find that the factorial is the largest as soon as $n > 8$. Without any approximation, the factorial is the largest as soon as $n > 2$ [$n=2$ is the only solution of $n! = n^{n/2}$]