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
These are the th-roots of the complex number .