Puzzled over basic exponentiation problem with complex argument

Question

I can't figure out where is my erroneous thinking in the following elementary use of exponents. I have used Mathematica to verify that I don't know what I'm doing.

$$ a^{bc}=\left(a^{b}\right)^{c} $$ Mathematica says "True".

$$ e^{bc}=\left(e^{b}\right)^{c} $$ Mathematica says "True".

$$ e^{ibc}=\left(e^{ib}\right)^{c} $$ Mathematica says "True" ($i=\sqrt{-1}$).

$$ e^{i2\pi c}=\left(e^{i2\pi}\right)^{c} $$ Mathematica says "False".

FWIW I used PowerExpand[] on each of these expressions in Mathematica and put them inside a TrueQ[] function. I know that $ e^{i2\pi} = 1 $, that $ 1^{c}=1 $, and that $ e^{i2\pi c} $ is generally some point on the unit circle depending on the value of $c$ and thus a complex number with unit magnitude.

Please show me what is wrong.

Answer 1

The documentation for PowerExpand clearly states:

The transformations made by PowerExpand are correct in general only if c is an integer or a and b are positive real numbers.

This is why you are getting erroneous results. In the last expression, if $c$ is not an integer, $e^{2\pi i c} \ne (e^{2\pi i})^c$; e.g., try $c = 1/2$.

Do not use Mathematica commands without first understanding the documentation.