You had better ask your instructor what exactly he/she means by $\sqrt[3]{\sqrt3-i}$.
I argue that such notation is ambiguous. Today's complex annalist do not like radicals applied to complex numbers.
You have gotten two replies. One assumes the notation asks about what is called the principal cube root.
The other seems to imply that it asks for the three cube roots (I may have missed the intent there)
But I hope you get my drift. Is your teacher asking for one answer or three answers?