Fallacy involving Euler's formula

Question

If $$ e^{ix} = \cos{x} + i \sin{x} $$ which means $$ e^{2\pi i} = \cos{2\pi} + i \sin{2\pi} = 1 + 0i = 1 $$ and $$ a^{(bc)} = (a^b)^c $$ then why is this wrong for some real number $x$? $$ e^{ix} = e^{2\pi i(x/2\pi)} \\ = (e^{2\pi i})^{(x/2\pi)} \\ = 1^{(x/2\pi)} \\ = 1 $$ ?

Answer 1

The "rule" $a^{bc}=(a^b)^c$ simply does not hold for complex numbers. Indeed, it doesn't even hold for all real numbers (consider $a=-1$ and $b=2$ and $c=\frac12$).

This is a great example of how, as mathematicians, we need to remember more than just the equations we learn—we need to remember the precise conditions under which the equations hold.