Find the fifth roots of 32 (cos 280° + i sin 280°).

I) Let z = 32 (cos 280 degrees + i sin 280 degrees)

= 32 (cos (k * 360 + 280) + isin (k*360 + 280)), where k = integer

II) z^ (1/5) = (32 (cos (k*360 + 280) + i sin (k*360 + 280))) ^ (1/5)

then,

z^ (1/5) = 2 (cos ((k*360 + 280) / 5) + i sin ((k*360 + 280) / 5))

We apply De Moivre's theorem:

z^ (1/5) = 2 (cos (72k + 56) + i sin (72k + 56))

We can get the five roots by assigning k = 0, 1, 2, 3, and 4

When K = 0, 1st root = 2 + i sin (cos 56 + i sin 56) k = 1, 2nd root = 2 (cos 128 + i sin 128) k = 2, 3rd root = 2 (cos 200 + i sin 200) k = 3, 4th root = 2 (cos 272 + i sin 272) k = 4, 5th root = 2 (cos 344 + i sin 344)

The fifth root is 5th root = 2 (cos 344 + i sin 344)