# If $z_1,z_2,z_3,z_4$ are the roots of $z^4=1$, then $\sum\limits_{i=1}^4 z_i^3$=?

$\begin{array}{1 1}(A)4 \\ (B) 0 \\ (C) 1+i \\(D) 1-i \end{array}$

Toolbox:
• All the four fourth roots of 1 are $1,-1,i,-i$
Given $z_1,z_2,z_3,z_4$ are fourth roots of 1
$\Rightarrow\:z_1=1,z_2=-1,z_3=i,z_4=-i$
$\Rightarrow\:\sum\limits_{i=1}^4 z_i^3=1^3+(-1)^3+(i)^3+(-i)^3$
$=1-1-i+i=0$