I have a serious problem.... with e

First, I lay down the following theorem:

e^(pi*i) = -1

Anyone will accept this as truth, and a calculator will verify it.

By laws of exponents:

(e^(pi))^i = -1

By definition:

i = sqrt(-1) = (-1)^(1/2)

So again by law of exponents:

((e^pi)^(-1))^(1/2) = -1

To help (parenthesis may be causing confusion), this is pronounced:

quantity e to the pi, to the negative one, to the one half, equals negative one.

Here's where the confusion begins. Square both sides.

( ((e^pi)^(-1))^(1/2) )^2 = ( -1 )^2
(e^pi)^(-1) = 1

As we know, anything to the negative first power is simply an inverse (for instance- 2^(-1) = 1/2)

1/(e^pi) = 1

e^pi is a constant.

2.71828^3.14159 = 23.14069

Therefore by Euler's formula:

e^(pi*i) = -1
1/23.14069 = 1

((e^pi)^(-1))^(1/2) = -1
moar like (e^pi)^((-1)^(1/2)) = -1 amirite

also a lot of laws are only valid for the reals, not for complex numbers. but i'm too lazy to look up whether you're using any of those.