Writing N-th roots of unity

Question

I have a question regarding roots of unity. In general, we can write the n-th roots of unity as $$e^{2\cdot\pi\cdot i\cdot\frac{k}{n}}$$.

However, if we do the following manipulation we get the following:

$$e^{2\cdot\pi\cdot i\cdot\frac{k}{n}}=(e^{2\pi\cdot i})^{k/n} = 1^{k/n}$$

The first part seems to me only apply the rules of exponentiation, the last part also seems reasonable logically, but makes no sense. Could anyone tell me what I am doing wrong here?

The rule $z^{ab} = (z^a)^b$ is generally not true when we are dealing with complex numbers. This has been answered many times before but I cannot find a good duplicate right now, hopefully someone else will find it. — Winther, Feb 01 '16 at 01:34
The second part has to make you realise that: $$\sqrt{1}=\pm1=1^{\frac12}$$So everything works out. — Simply Beautiful Art, Feb 01 '16 at 01:39
Similar post: http://math.stackexchange.com/questions/656198/why-the-square-root-of-x-equals-x-to-the-one-half-power/656209#656209 — Simply Beautiful Art, Feb 01 '16 at 01:40
Wikipedia has a section on this. Exponentiation: Failure of power and logarithm identities — Winther, Feb 01 '16 at 01:41

score 0 · Answer 1 · answered Feb 01 '16 at 01:38

You are kind of right and kind of wrong.

See, if I had:

$$1^i\ne1$$

Then you might realise that $1^x\ne1$ for $x\in\mathbb{C}$. However, it is a) not obvious and b) gives rise to an infinite amount of branches.

For $k,n\in\mathbb{R}$, we have the following:

$$|e^{2\pi i\frac kn}|=1$$

Verifying your idea. But again, this is only for real $k,n$.

Writing N-th roots of unity

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