Question

Find the fifth roots of 243(cos 240° + i sin 240°)

Answer 1

The best way to find a root is to get the root of the coefficient, then divide the angle by what root you want for the first one, then add 360/root to get each additional angle until you have all the roots. So for the fifth root, get the fifth root of 243, then divide 240 by 5 for your first angle. You then write the root in the same form as the original, just with your new numbers. Now take 360/5 to find that you will add 72 to the angle to get your next angle. Keep adding 72 to get another angle until you have five roots in all. 

Answer 2

Apply De Moivre's Theorem

zⁿ = rⁿ(cos.n.Ф + i.sinn.Ф).

The 5th root, that means that n = 1/5  ↔↔↔ rⁿ = r¹/₅ =    ⁵√r

zⁿ = ⁵√(243) [cos (240+2kπ)/5 +i.sin(240+2kπ)/5]
with k = 1,2,3,..,n-1. Note that ⁵√(243) = 3

1st root : for k=0 → 3[cos(240/5)+isin(240/5) = 3(cos48°+isin48°)
2nd root : for k=1→3[cos(240+360)/5)+isin(240+360)/5]=3(cos120°+isin120°)
3rd root : for k=2 →→ 3(cos192 + isin192)
4th root : for k=3→→3(cos264+isin264)
5th root : for k =4↔↔3(cos336+isin336)