More specifically, what is $e^i$ (exact value)? Can I use regular exponent rules for imaginary numbers? Do regular exponent rules apply to both (like a complex number)?
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3$e^{i\theta} = \cos(\theta)+i\sin(\theta)$ so $e^i = \cos(1)+i\sin(1)$ (N.B: this is written with radians) – JMoravitz Sep 28 '20 at 18:03
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As for your other questions, which are not answered on the other thread: yes and no. For any real $n > 0$, we can define $$n^{(a+bi)} = n^ae^{ib\ln n} = n^a(\cos (b\ln n) + i\sin(b\ln n))$$ which satisfies $n^{(z + w)} =n^zn^w$ and $(nm)^z = n^zm^z$ for any complex $z, w$ and real $n, m > 0$. But when the base $n$ is complex or negative, there are multiple values for $\ln n$, called branches, and you get different behavior depending on which branch you choose. The exponential rules still hold in the sense that you can choose branches on each side that make them true. – Paul Sinclair Sep 29 '20 at 01:41