$4 \cdot 36$ Try to find the $n$ $n$-th roots of 1, and find the sum of their $n$-th powers.

 untranslated part:
 
$4 \cdot 36$ 试求 1 的 $n$ 个 $n$ 次方根, 并求它们 $n$ 次幂的和.
[Solution] $x^{n}=1$, that is, $x^{n}=\cos 2 k \pi+i \sin 2 k \pi \cdot(k=0, \pm 1, \pm 2, \cdots)$, so by De Moivre's theorem, the $n$ roots are:
$$
x_{k}=\cos \frac{2 k \pi}{n}+i \sin \frac{2 k \pi}{n} \cdot(k=0,1,2, \cdots, n-1)
$$

Since $\quad x_{k}^{n}=1 .(k=0,1,2, \cdots, n-1)$
Therefore, $x_{0}^{n}+x_{1}^{n}+\cdots+x_{n-1}^{n}=\underbrace{1+\cdots+1}_{n \uparrow}=n$.