Question

If the fourth roots of unity are $z_1,z_2,z_3,z_4$ then $z_1^2+z_2^2+z_3^2+z_4^2$ is equal to

Answer 1

Using the $n$th root formule we get
$1^{1/4}=e^{i\left(2k\pi /4\right)};k=0,1,2,3\Rightarrow 1^{1/4}=e^{0i},e^{i\left(\pi /2\right)},e^{i\left(\pi \right)},e^{i\left(3\pi /2\right)}$
$\therefore 1^{1/4}=1,i,-1,-i$
$\therefore z_1^2+z_2^2+z_3^2+z_4^2=1^2+i^2+(-1{)}^2+(-i{)}^2=0$