# z = ( -125/2(1+sqrt{3} i))^(1/3) #

#= (-125)^{1/3} (1/2 + sqrt{3}/2 i )^{1/3}#

#= -5 (cos pi/3 + i sin pi/3)^{1/3}#

#= -5 (cos (pi/3+ 2pi k) + i sin (pi/3 + 2 pi k) )^{1/3} quad# integer #k#

We add the #2pi k# because every complex number has three cube roots and we want to find them all.

De Moivre's theorem works when #n# is a fraction too; it's basically Euler's Formula.

#= -5 (cos (pi/9 + {2pi k}/3) + i sin (pi/9 + {2pi k}/3))#

That's three unique values, given by any three consecutive #k.# These aren't constructible angles so there's no nice expression combining integers, addition, subtraction, multiplication, division and square rooting. We'll just write the three possibilities, #k=-1,0,1.#

#z = -5 (cos (-{5pi}/9) + i sin (-{5pi}/9))# or

#z = -5 (cos (pi/9) + i sin (5pi/9))# or

#z = -5 (cos ({7pi}/9) + i sin ({7pi}/9))#

We can lose the minus sign by adding #pi# to the angles, but I won't bother.