Question

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

Answer 1

See below.

Explanation:

Fifth root of 243 = 3,

Suppose r( cos Ф + i sinФ) is the fifth root of 243(cos 240 + i sin 240),

then r^5( cos Ф + i sin Ф )^5 = 243(cos 240 + i sin 240).

Equating equal parts and using de Moivre's theorem:

r^5 =243 and cos 5Ф + i sin 5Ф = cos 240 + i sin 240

r = 3 and 5Ф = 240 +360p so Ф = 48 + 72p

So Ф = 48, 120, 192, 264, 336 for 48 ≤ Ф < 360

So there are 5 distinct solutions given by:

3(cos 48 + i sin 48),

3(cos 120 + i sin 120),

3(cos 192 + i sin 192),

3(cos 264 + i sin 264),

3(cos 336 + i sin 336).. (Answer).