How would you simplify:

2^pie-2
----------
2^pie+2

Do you mean 2^(pi-2)/2(pi+2)?

That can be written
2^(pi -2 -pi -2)= 2^-4 = 1/16.

If you mean [(2^pi)-2]/[(2^pi)+2],
that is about as simplifierd as you can make it.