Hi i need help for calcul:

Question

( 1/2 ( -1 + i√3 )) ^3
and
( 1/2 ( -1 - i√3 )) ^3
thanks for your help

Answer 1

use de Moivre's theorem

= (1/8) (-1+√3i)^3

let's just concentrate on the -1+√3
magnitude = √4 = 2
so
-1 + √3 = 2(cosØ + isinØ)
where tanØ = -√3/1 , Ø = 120°

so -1 + √3 = 2(cos120 + i sin120°)
then by De Moivre's theorem

(-1+√3)^3 = 2^3 (cos 360 + i sin360)
= 8(1 +0)
= 8

so (1/8)(-1+i√3)^3
= (1/8)(8)
= 1

do the same with your second question, (your angle will be
240°)