problemo....cube roots of unito

Ans1) The eqn zⁿ-1 has n roots which are called the nth roots of unity.
Zn = 1 = cos0 + i sin0
= cos 2k∩ + i sin 2k∩
Therefore, Z = cos 2k∩ / n + i sin sin 2k∩ / n where k=0,12,3,......, n-1
Zⁿ-1 = (z-1)(z^n-1 + z^n-2+ ...........+z+1)

Z ⁿ - 1 =

(zⁿ-1) = (z-1)

or, (z n-1) / (z-1) = ..........(1)

Put n=5 and z=-w, then
(-w⁵-1) / (-w-1) =

(w²+1) / (w+1) =

or -w / -w² =

= 1/w -------------------(2)

Also in (1), put z=w², n=8, r=q, then

(w¹⁶-1) / (w²-1) =

1 / (w+1) =

-1/ w² = .......................(3)

From (2) and (3),
/ = -w
=(1- i √3) / 2

The answer is absolutely rite...............