I found the following 2 questions regarding e and pi in the special topics of my precalculus math book.
(1) e^(pi) + pi^(e) =
(2) e^(pi)^(e) + pi^(e)^(pi) =
There is nothing special about them. I suspect that they are just exercises in using your calculator. Probably your calculator has a "[itex]\pi[/itex]" key. If not, you can always use [itex]\pi= 4tan^{-1}(1)[/itex] (with your calculator in radian mode, of course). For "e" use either the " key to find or use the "inverse" "ln" keys.
No, unless have have an integrated floating point unit processor in your cerebral cortex. And even with infinite processing power, you would not be able to get the exact answer, since and are irrationnal. Thus, you must use a calculator to get the best "approached" value.