The sum of 162 ^ th power of the roots of the equation x ^ 3 - 2 x...

${x}^{3} - 2{x}^{2} + 2x - 1 = 0$

$x = 1$ satisfying the equation

$∴ x - 1$ is factor of ${x}^{3} - 2{x}^{2} + 2x - 1$

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

$x = 1, \dfrac{1 + i\sqrt{3}}{2}, \dfrac{1 - i\sqrt{3}}{2}$

$x = 1, - {\omega}^{2}, - \omega$

sum of ${162}^{th}$ power of roots

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

$= 1 + {\left(\omega\right)}^{324} + {\left(\omega\right)}^{162}$

$= 1 + 1 + 1 = 3$