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I assumed I could use this identity for any number but I figured it doesn't always work once I tried it with complex numbers.

For example, consider $e^{2aπi}$, with $a \epsilon \mathbb{Q}$

According to Euler's formula, $e^{2\pi ai}=\cos{(2\pi a)}+i\sin{(2\pi a)}$, which gives a complex number that depends on the rational number $a$.

However, using the property $ {(a^{b})}^{c}=a^{bc}$, we get:

$e^{2\pi ai}={(e^{2\pi i})}^a=1^a=1$

SBL
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  • In general when you define the function $z \mapsto z^a$ for complex numbers, you have to choose a branch. We define $z^a = \exp(a \ln z)$, and the logarithm is multivalued, which is where the need for a choice of branch comes from. – Haydn Gwyn Jan 01 '23 at 18:42
  • There are many questions on this site about that kind of paradox. Does this help: https://math.stackexchange.com/questions/4412683/proofs-request-proofs-that-five-exponention-rules-hold-for-positive-real-bases/4412691#4412691 – Ethan Bolker Jan 01 '23 at 18:52
  • Ethan Bolker, I don't have a proof that the property works for all a,b,c in C, so I haven't found a counterexample to something proven for there to be a paradox. What I am asking is up to which level of numbers can we go where there is a clear proof the property holds before the property stops holding beyond that. – SBL Jan 01 '23 at 20:16
  • An entirely tangent point on sentence interpretation: the broadest possible restrictions would be something like "when $a = b = c = 1$" - at least, if you want the statement to still have some meaning. But everyone knows what you really mean, so feel free to ignore this. – Paul Sinclair Jan 02 '23 at 17:42

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