Question

If -1,$\alpha$,${\alpha }^3$,${\alpha }^5$,$\overline{\alpha }$,$\overline{{\alpha }^3}$,$\overline{{\alpha }^5}$ are roots of the equation $z^7$ + 1 = 0.Find the value of $\alpha$$\overline{\alpha }$,${\alpha }^3$$\overline{{\alpha }^3}$,${\alpha }^5$$\overline{{\alpha }^5}$,$\alpha$$\overline{\alpha }$ + ${\alpha }^3$$\overline{{\alpha }^3}$ + ${\alpha }^5$$\overline{{\alpha }^5}$.

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Answer 1

$z^7$ = -1

z = $(-1{)}^{\frac{1}{7}}$c

= $(cos\pi +isin\pi {)}^{\frac{1}{7}}$

= $cos\left(\frac{\pi +2k\pi }{7}\right)+isin\left(\frac{\pi +2k\pi }{7}\right)$-----------------(1)

Where k = 0,1,2,3,4,5,6

$\alpha$.$\overline{\alpha }$ = $|a{|}^2$ = 1 {From equation 1 we see that modulus of each root is 1}

${\alpha }^3$.$\overline{{\alpha }^3}$ = $|a^3{|}^2$ = 1

${\alpha }^5$.$\overline{{\alpha }^5}$ = $|a^5{|}^2$ = 1

$\alpha$$\overline{\alpha }$ + ${\alpha }^3$$\overline{{\alpha }^3}$ + ${\alpha }^5$$\overline{{\alpha }^5}$.= 1 + 1 + 1 =3