Let z = 81 (cos100° + i sin100°). Find all of the 4th roots of z. Write the answers in trigonometric form

Question

Let z = 81 (cos100° + i sin100°). Find all of the 4th roots of z. Write the answers in trigonometric form

Answer 1

z^(1/4) = (81^.25)(cos100/4 + isin100/4) = +/-3(cos25+isin25), +/-3i(cos25+isin25)

81 has 4 roots, -3,+3, -3i, +3i

Answer 2

z = 81(cos100° + isin100°)

There are 4 fourth roots of z.

First 4^th root: 81^1/4[cos(100°/4) + isin(100°/4)] = 3{cos25° + isin25°)

Second 4^th root: 3[cos(25°+ 360°/4) + isin(25°+360°/4)] = 3(cos115° + isin115°)

Third 4^th root: 3[cos(25° + 2(360°/4) + isin(25° + 2(360°/4)] = 3(cos205° + isin205°)

Fourth 4^th root: 3[cos(25° + 3(360°/4 ) + isin(25° + 3(360°/4)] = 3(cos295° + isin295°)