Roots of unity notation

The question I'm about to ask sounds furiously idiotic, but it's been driving me nuts so here goes. Recall Euler's formula, $$e^{i\theta} = \cos\theta + i\sin\theta.$$ In particular, $e^{2\pi i} = 1$. Now, for a positive integer $n$, the $n^{th}$ roots of unity are denoted $$(\zeta_n)^k\quad\text{where}\quad\zeta = e^{2\pi i/n},\quad k = 1,\dots ,n.$$ My question is: why? Wouldn't it make more sense to instead denote the first $n^{th}$ root of unity by $1^{1/n}?$

• Consider $n=2$. What is $1^{1/2}$? Could be $1$; could be $-1$. (This can be covered by introducing an additional convention around notation, yes...) But now consider $e^{2\pi i}$. This is, as you note, $1$, and without ambiguity. – Benjamin Dickman Mar 31 '17 at 23:10

For example, what would $2^{1/n}$ mean in your notation? While one $n$th root of unity generates the rest, one value for $2^{1/n}$ does not generate the rest of the values for $2^{1/n}$. It would be odd to use $1^{1/n}$ to mean a complex root of unity, and $2^{1/n}$ to mean a real root.