raising a complex number to a high power.

Question

we should decide whether the following claims are right or not, and explain our decision.

let $w_1,w_2,w_3$ be three different roots for the equation $z^3=1$

a) $w_1^{1991} + w_2^{1991} + w_3^{1991}=1$

b) $w_1^{1991} + w_2^{1991} + w_3^{1991}=0$

I found that:

$w_1$=$1$,

$w_2$=${-1\over 2}$+i$\sqrt{3}\over 2$

$w_3$=${-1\over 2}$-i$\sqrt{3}\over 2$

Now, how do I continue from here? how dow I know what happens when I raise a complex number to very high power?

Answer 1

Hint: If $z^3=1$, then e.g. $z^{29}=z^{27+2}=z^{27}\cdot z^2=z^2$.