1 complex no.Q

w=e^(i * 2pi /3)
z=x^3

Let z=re^(iθ)

roots of x are defined as:
x_k= r^(1/3)* e^(i*(θ+2kpi)/3)

x_(k+1)= r^(1/3)* e^(i*(θ+2(k+1)pi)/3)
= r^(1/3)* e^(i*(θ+2kpi)/3+i*2kpi/3)

i.e. x_(k+1)= r^(1/3)* e^(i*(θ+2kpi)/3) * e^(i*2kpi/3) = x_k*w

x_(k+2)= r^(1/3)* e^(i*(θ+2(k+2)pi)/3)
=r^(1/3)* e^(i*(θ+2kpi)/3+i*2kpi/3 +i*2kpi/3 )

i.e x_(k+2)== x_k*w *w = x_k *w^2


it is true for all k
whatever which root you obtained, you can also find other roots by mult. w and w^2


PS. the equation must be in form of "z=x^3"

this is generally accepted fact in math.
you don't need to prove
If you want to learn more on "w", find "root of unity"