  For a given positive integers $m$ and $\ell$, we give a complete list of
positive integers $n$ for which their exist $m$th roots of unity $x_1,\dots,x_n
\in \mathbb{C}$ such that $x_1^{\ell} + \cdots + x_n^{\ell}=0$. This extends
the earlier result of Lam and Leung on vanishing sums of roots of unity.
Furthermore, we prove that for which integers $n$ with $2 \leq n \leq m$, there
are distinct $m$th roots of unity $x_1,\dots,x_n \in \mathbb{C}$ such that
$x_1^{\ell} + \cdots + x_n^{\ell}=0$.
