1

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?

Firas Abd El Gani
  • 2,136
  • 1
    Your two statements contradict each other. Anyhow, use the fact that $w_1^3=w_2^3=w_3^3$ to reduce the power of each to either $1$ or $2$. – Quang Hoang Oct 27 '14 at 20:04

1 Answers1

3

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

Berci
  • 90,745
  • this is brilliant, didn't think about it that way, that means, if w1,w2,w3 are roots, it means that not only the multiplication of all of them must give 1, but also $w_1^3$=1 , $w_2^3$=1 , $w_3^3$=1, right? – Firas Abd El Gani Oct 27 '14 at 20:37
  • It means exactly that $w_i^3=1$ for each $i$, yes. – Berci Oct 27 '14 at 22:33