How is the principal argument of the root of a complex number found where n is not an integer?

Question

Consider the complex number z = 1 − 6i

I am trying to compute the following:

• The modulus: |z|= sqrt(37)

• The principal argument: arg(z) = arctan(−6)

There are four distinct fourth roots to this number. I need to compute the following:

• The modulus of the fourth roots: |z^(1/4)|= 37^(1/8)

The principal arguments of the fourth roots of z are, in increasing order:

Smallest argument:

For this I tried to use arg(z^n) = n arg(z)

But on wikipedia it states that this can't be used for a non-integer value of n when working with principal arguments. How can I solve this?

https://en.wikipedia.org/wiki/Argument_(complex_analysis)#Identities

Also how do I get the next largest arguments?

next largest argument =

next largest argument =

largest argument =