you can prove it as follows:
You draw a graph with e^x and x^e. At x=e, both graph touch/cross.
With first deviation you find out that they touch. so they rise at same rate in this point. But as
1.) x^e (which represents the graph on which pi^e is situated at x=pi in this case) is a function where the exponent stays the same all the time, and
2.) e^x (representing e^pi for x=pi) where the exponent rises as x rises, the function e^x will have an increased gradient, thus be higher than x^e when x>e. As x=pi in our problem, and pi>e, we can say that e^pi is a larger number than pi^e.