Question

Which expression is a fourth root of -1+isqrt3?
a) fourth root of 2(cos(280degrees)+isin(280degrees))
B) fourth root of 2 (cos(30degrees)+isin(30degrees))
C) fourth root of 2 (cos60degrees)+isin(60degrees)
D) fourth root of 2( cos320degrees)+isin(320degrees))

Answer 1

\[(-1 + i \sqrt{3})^{1/4} = [2(\cos 120 + i \sin 120)]^{1/4}\]
\[ = \sqrt[4]{2} ( \cos \frac{120 +360k}{4} + i \sin \frac{120+360k}{4})\]
\[k = 0,1,2,3\]
---> plug in each k value to get 4 roots
Only one will match given answers