Solution
= Solution
The required <root of unity> is
$$
\omega=e^{2\pi i/3}=-\frac12+\frac{\sqrt3}{2}i,
$$
so
$$
1+\omega=\frac12+\frac{\sqrt3}{2}i=e^{i\pi/3}.
$$
Therefore
$$
(1+\omega)^{10}=e^{10\pi i/3}=e^{4\pi i/3}
=\boxed{-\frac12-\frac{\sqrt3}{2}i}.
$$