# The sum of

Question:

The sum of $162^{\text {th }}$ power of the roots of the equation $x^{3}-2 x^{2}+2 x-1=0$ is

Solution:

Let roots of $x^{3}-2 x^{2}+2 x-1=0$ are $\alpha, \beta, \gamma$

$(x-1)\left(x^{2}-x+1\right)=0$

Now $\alpha^{162}+\beta^{162}+\gamma^{162}$

$=1+\omega^{162}+\left(\omega^{2}\right)^{162}$

$=1+\left(\omega^{3}\right)^{54}+\left(\omega^{3}\right)^{108}$

$=3$