Question

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

Answer 1

Answer: 3

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