Applying property (2) of exponents, we have
Thus, the only thing new is . Since
we see that is the th root of .
This is sometimes written
The th root of a real (or complex) number is not unique. As we all know, square roots give two values (e.g., ). In the general case of th roots, there are distinct values, in general.
How do we come up with different numbers which when raised to the th power will yield ? The answer is to consider complexnumbers in polar form. By Euler’s Identity, the real number can be expressed, for any integer , as . Using this form for , the number can be written as
We can now see that we get a different complex number for each . When , we get the same thing as when . When , we get the same thing as when , and so on, so there are only cases using this construct. The th root for can be called the “primitive th root of ”, since integer powers of it give all of the others.