Perhaps we are all familiar with the square root , which in this term would be the 2nd root, and that with the result we get, by squaring it will give us . Taking the nth root of  is similar, but it gives us a result to which we need to increase it to the power of n to get back . Thus,

which should be easier for the reader to understand. Now relating it to complex numbers, when we are finding the nth root of complex number , we are essential finding a number such that by increasing it to the power of n, we get back .

We shall now formally state the principle.

Suppose that  is a positive integer and  is a given complex number. There are  distinct th roots of, which are defined by

for each .

Now by simply looking at this formula, we should inspect on some of its features. First, when we want to find the nth roots of , we will get n roots, or simply put, when finding the 5th roots of , we will get five different numbers that when each are individually taken to the power of five, we get back . Quite interesting I must add because it seems that there are five numbers we can use to get ‘back’ to  by multiplying with itself. Second, the nth roots are distinct so no two are the same. Third and quite importantly, the roots can be real or imaginary as a value of k might give sin=0. I hope you see that.

To show this principle, we first write  and  in polar form, which is

 and

where  and . By definition,

paying close attention to the application of de Moivre’s theorem on the last step. By equating the magnitudes and argument, we have

and

Recognizing the periodicity of the cosine function. This gives  distinct th roots of .

Now, must of you might think why did we set the range of values that k can take to be from 0 to n-1. Well, to show that, we need to consider the geometric representation of these nth roots, which is the next lesson.