Laurent expansion of principal root

How do I find the Laurent expansion of a function containing the principal branch cut of the nth root?

Example:
[tex]f(z)=-iz\cdot\sqrt[4]{1-\frac{1}{z^{4}}}[/tex]

 

Around any point a with |a|>1.

As far as I know, the principal nth root of z is defined as [tex]\sqrt[n]{|z|}e^{\frac{1}{n}i\mathrm{Arg}z}[/tex], which doesn't seem very helpful; how would you expand [tex]\sqrt{z}[/tex] around a point where it is defined?

Perhaps these are silly questions; my book is very vague on Laurent expansions...

EDIT: I think I've got it... or maybe not...