(-1)^(1/3)

Q: calculate (-1)^(1/3), here ^ is power operation. You are given normal mathematical functions: + - * / log exp cos sin

x^y = exp( log(x^y)) = exp( y * log(x))

x is negative, we need to know log(-1)

log(-1) = log( i^2) = 2 log(i)

let's assume

log( i) = i*a
i = exp (log(i)) = exp (i*a) = cos(a) + i*sin(a)
sin(a) = 1, cos(a) = 0
a = 2*k*pi + pi/2


log(-1) = 2log(i) = 2*i*a = (pi + 4*k*pi)*i


x^y = exp( 1/3 * log(-1))

The possible answers:
k=0,  x^y = exp( 1/3 * pi * i) = cos(pi/3) + i*sin(pi/3)
k=1,  x^y = exp( (pi + 4*pi)*i/3) = cos( 5*pi/3) + i * sin( 5*pi/3)
k=2,  x^y = exp( 9*pi*i/3) = cos(pi) + i*sin(pi) = -1


----- I should be much more simple -----
x=-1,  x=e^(i*pi)=e^(i*3*pi)=e^(i*5*pi)
answer is
x^(1/3) = e^(i*1/3*pi), e^(i*5/3*pi), e^(i*pi)

Interesting thing is, Matlab will only list one answer.