Which is larger $(101!)^{100}$ or $(100!)^{101}$ I am supposed to tell which one of $(101!)^{100}$ and $(100!)^{101}$ is larger. I am trying to use the behavior of the function $f(x)=x^{1/x}$ as is a standard technique to dealing with questions of this sort. Here is what I have so far.
$$\begin{aligned}(101!)^{100!}&\lt (100!)^{101!}\\ (101!)^{100} &\lt (100!)^{101\times 100}\end{aligned}$$
Any ideas on how to proceed. Thanks.

Well
$$(101!)^{100}\cdot (101!)=\color{green}{(101!)^{101}}$$
while:
$$(100!)^{101}\cdot (101)^{101}=\color{green}{(101!)^{101}}$$
the one you have to multiply by the larger number is smaller
