Question

This is from complex numbers please help​

Answer 1

Step-by-step explanation:

We have,

The given equation is

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

$\implies x = \frac{ - ( - 1) \pm \sqrt{( - 1)^{2} - 4(1)(1)} }{2}$

$\implies x = \frac{ 1 \pm \sqrt{1 - 4} }{2}$

$\implies x = \frac{ 1 \pm \sqrt{ - 3} }{2}$

$\implies x = \frac{ 1 \pm i\sqrt{ 3} }{2}$

So, roots are

$z_{1} = \frac{1 + i \sqrt{3} }{2} \\ z _{2} = \frac{1 - i \sqrt{3} }{2}$

But, we know, cube roots of unity are

$w = \frac{ - 1 + i \sqrt{3} }{2} \\ {w}^{2} = \frac{ - 1 - i \sqrt{3} }{2}$

So,

on comparing

we get,

$z_{1} = - {w}^{2} \\ z _{2} = - w$

So,

$( z_{1})^{2000} + ( z_{2})^{2000}$

$= ( - {w}^{2} )^{2000} + ( - w)^{2000}$

$= {w}^{4000} + w^{2000}$

$= {(w^{3} )}^{1333}.w + (w ^{3}) ^{666}. {w}^{2}$

$= {(1)}^{1333}.w + (1) ^{666}. {w}^{2}$

$= w + {w}^{2}$

$= - 1$