First we need to write z in its polar form, i.e in the form r(cosq+ isinq).
To do this, we first find q, and to do that we use the fact that y/x=tanq.
In this case, y/x is minus the square root of 3, so we need to remember (or look up) which angle has tanq equal to minus the square root of 3.
It's q=120o.
Now that we know q, we can work out the modulus r. For this we use the fact that x=rcosq, where x is the real part of our original complex number z.
In our case the real part of z is -1, so x=-1. Then since x=rcosq, that means rcosq must be -1.
We know that q=120o, so cosq=-1/2. So r must be 2, to make rcosq=-1.
In polar form our complex number is therefore:
That's most of the work done now. All we have left to do is to take the cube root of z, which means taking the cube root of the modulus and multiplying the angle by 1/3:
We can then return this to Cartesian form again, by calculating the cube root of 2 and the cosine and sine of 40o:
