Example 10. If $1+x+x^{2}+x^{3}+x^{4}+x^{5}=0$, then $x^{6}=$ $\qquad$ .
From the given conditions, we know that $x \neq 1$. Therefore,
$$
(x-1)\left(x^{5}+x^{4}+x^{3}+x^{2}+x+i\right)=0 \text {. }
$$
$x^{6}-1=0$, i.e., $x^{6}=1$