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

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.

Apply De Moivre's Theorem

z ⁿ = 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)