As mentioned in §3.4, there are different numbers
which satisfy when is a positive integer.
That is, the th root of , which is
written as , is not unique--there are of them. How do
we find them all? The answer is to consider complex numbers in
polar form.
By Euler's Identity, which we just proved, any number,
real or complex, can be written in polar form as

where and
are real numbers.
Since, by Euler's identity,
for every integer , we also have

Taking the th root gives

There are different results obtainable using different values of
, e.g.,
. When , we get the same thing as
when . When , we get the same thing as when , and so
on, so there are only distinct cases. Thus, we may define the
th th-root of
as