How do you find the fourth root of #-8 + 8sqrt3i#?

1 Answer
Jan 2, 2017

#-8+8sqrt(3)i# has four fourth roots. These are #sqrt(3)+i#, #1-sqrt(3)i#, #-sqrt(3)-i# and #-1+sqrt(3)i#.

Explanation:

Plot #-8+8 sqrt(3)i# on the complex plane and note that it has an argument of #120°# and a modulus of #sqrt((-8)^2+(8 sqrt(3))^2)=16#.
Consequently the argument and modulus of one root are #(120°)/4# and #16^(1/4)#, that is, #30°# and #2#, which is #sqrt(3)+i#. The other three have the same modulus and arguments which are multiples of #90°# added on.