# Fallacy involving Euler's formula [duplicate]

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 $$x$$? $$e^{ix} = e^{2\pi i(x/2\pi)} \\ = (e^{2\pi i})^{(x/2\pi)} \\ = 1^{(x/2\pi)} \\ = 1$$ ?

• Cf. this question: fractional powers of negative numbers are not uniquely defined, and the "general rule" $(a^b)^c=a^{b×c}$ does not always work when $b$ and $c$ are not integers – J. W. Tanner Dec 25 '20 at 20:03

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 $$c=\frac12$$).
• So when can I count on the rule being true? I assume if $a$ is a positive real and $b$ and $c$ are real. Also if $b$ and $c$ are both integers, right? When else? What if one of $b$ or $c$ is an integer? – ajb Dec 26 '20 at 4:27