Proving Sum of Roots is Zero The question is if $w$ is any real number , the equation $z^k = w$ has $k$ roots, 
$k ∈ n$.
Show that the sum of the k roots is zero.

Usually with these root questions I convert the right hand side to polar form so in this case
$z^k = w cis 0$
$z = w^{1/k} cis 1/k$ $(0 + 2πn)$ (n is any integer) 
$z = w^{1/k} cis 2πn/k$ 
My thought was writing $k$ as $1,2,3,4,5$ and so on but that would be too long , is there an easier method of solving this?

If $r_1, r_2, \dots, r_k$ are the roots of the polynomial $z^k-w$, then $$z^k - w = (z-r_1)(z-r_2)\cdots(z-r_k)$$
Assuming $k>1$, the coefficient of $z^{k-1}$ on the left side is $0$, while on the right side, it is $-r_1-r_2-...-r_k$, so:
$$0=-r_1-r_2-...-r_k $$
and hence
$$0=r_1+r_2+...+r_k$$
