Everyone knows that e^[pi(i)] == -1

Using this we also know that e^[2pi(i)] == 1

What I believe I proved was e^i == 1 using the following proof:

Start with [e^i]

Since raising a number to 1 just gives you the number let's raise this to the equivalent of 1 or [(2pi)/(2pi)]

[e^i]^[(2pi)/(2pi)]

When raising a power to a power you just multiply them to get the new power also when multiplying 2 number you can also pull them apart as follows

i*[(2pi)/(2pi)] == [2pi(i)]*[1/2pi] returning to the original problem we now get

[e^(2pi(i))]^[1/(2pi)] but since we already know that [e^(2pi(i))] == 1 we get the following

[1^(1/{2pi})] but 1 raised to any power equal to 1 therefore we have e^1 eqaul to 1 which we know isn't correct.

Where did I mess up?