In looking for the fifth roots, we will use De Moivre's theorem.
The formula for this problem is z^5 = 243 (cos 300 degress + i sin 300 degrees)
Where you'll also need the following data:
300/5 = 60
360/5 = 72
- you'll input these two after the cos and i sin (60+k*72) where k = 0,1,2,3,4
solution:
z^5 = 243 (cos 300 degrees + i sin 300 degrees)
z= 243^1/5 (cos 300 degrees + i sin 300 degrees)
z= 3 (cos (60 + k*72) degrees) + (i sin (60 + k*72) degrees)
so the following are the roots:
3 (cos 60 degrees + i sin 60 degrees)
3 (cos 132 degrees + i sin 132 degrees)
3 (cos 204 degrees + i sin 204 degrees)
3 (cos 276 degrees + i sin 276 degrees)
3 (cos 348 degrees + i sin 348 degrees)