expressing z = 1 + √3i in mod-arg form gives us 2.cis(π/3) so,
z<sup>n</sup> = 2<sup>n</sup>cis(nπ/3)
z<sup>n</sup> is real when Im(z<sup>n</sup>) = 0 , that is, when Im(z<sup>n</sup>) = 2<sup>n</sup>sin(nπ/3) which is equal to zero when nπ/3 = kπ ----> n=3k where k =...