Exponent identities with imaginary exponents$\left(a^i\right)^i$

Question

I've been trying to understand how imaginary exponents work, and I think I mostly understand it, but I'm confused by something like $\left(a^i\right)^i$ (where $a$ is real). According to the normal exponent identities, this should be equivalent to $a^{i\cdot i} = a^{-1} = \frac 1 a$, but since $a^i$ is understood as $e^{i \ln a}$ which is rotating $\ln a$ radians around the unit circle, it's clear that that identity isn't going to hold forever, because eventually as $a$ increases, it's going to wrap all the way around the cicle, whereas $\frac 1 a$ doesn't wrap around in any sense. And indeed, python tells me that $\left(24^i\right)^i \approx 22.312$, which is nowhere near $\frac 1 {24} \approx 0.4167$. So what am I missing?

Answer 1

$(a^b)^c=a^{bc}$ is not valid when $b$ has imaginary part otherwise one would have

$$e^{-4\pi^2}=(e^{2i\pi})^{2i\pi}=1^{2i\pi}=1$$

And this is absurd