Let be a nonzero complex number where . Then, let and and define another complex number . Now, let for .
- Prove that . We say is an th root of .
- Prove that has exactly distinct th roots given by
and that they are equally spaced on a circle of radius .
- Find the three cube roots of .
- Find the four fourth roots of .
- Find the four fourth roots of .
- Proof. Using the definitions of and we compute,
- Proof. From part (a) we know is an th root of . Then, if we have
(since for all ). Hence, is an th root of .
Then, if and we have for . This implies
for ; hence, there are distinct values of . Therefore, has exactly distinct th roots. By the Fundamental Theorem of Algebra we know it cannot have more than roots; hence, we have shown that they are all of the form
- For the complex number we have
Therefore, from parts (a) and (b) we have
Hence, the three cube roots of are
- This time we have
Therefore the four fourth roots of are
- For we have
Therefore,
Thus,