Question

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

Answer 1

$x^{3}-2 x^{2}+2 x-1=0$
$x =1$ satisfying the equation
$\therefore x-1$ is factor of
$x^{3}-2 x^{2}+2 x-1$
$=(x-1)\left(x^{2}-x+1\right)=0$
$x=1, \frac{1+i \sqrt{3}}{2}, \frac{1-i \sqrt{3}}{2}$
$x =1,-\omega^{2},-\omega$
sum of $162^{\text {th }}$ power of roots
$=(1)^{162}+\left(-\omega^{2}\right)^{162}+(-\omega)^{162}$
$=1+(\omega)^{324}+(\omega)^{162}$
$=1+1+1=3$