It is a known fact that
and in general a square root always has two roots, one positive and one negative. Also
has four possible answers, since
Briefly, if with take the nth root of a number we obtain n answers: This is illustrated on the Argand diagram below which shows ten tenth roots of 1.
To find the nth roots of a complex number we write it in polar form if it is not already in polar form. This done we apply the following:
where
Example Find the cube roots of 27
so the cube roots are given by:
with
in order
ie the cube roots are
The last simplifies so the roots are
Example Find the 4 th roots of 16i
so the cube roots are given by:
with
in order
ie the fourth roots are