# Comparing \pi^e\pi^e and e^\pie^\pi without calculating them

How can I

compare (without calculator or similar device) the values of $$\pi^e\pi^e$$ and $$e^\pie^\pi$$ ?

Another proof uses the fact that $$\displaystyle \pi \ne e\displaystyle \pi \ne e$$ and that $$e^x > 1 + xe^x > 1 + x$$ for $$x \ne 0x \ne 0$$.

We have $$e^{\pi/e -1} > \pi/e,e^{\pi/e -1} > \pi/e,$$

and so

$$e^{\pi/e} > \pi.e^{\pi/e} > \pi.$$

Thus,

$$e^{\pi} > \pi^e.e^{\pi} > \pi^e.$$

Note: This proof is not specific to $$\pi\pi$$.