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# Prove some properties of nth roots of complex numbers

Let be a nonzero complex number where . Then, let and and define another complex number . Now, let for .

1. Prove that . We say is an th root of .
2. Prove that has exactly distinct th roots given by and that they are equally spaced on a circle of radius .

3. Find the three cube roots of .
4. Find the four fourth roots of .
5. Find the four fourth roots of .

1. Proof. Using the definitions of and we compute, 2. 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 3. For the complex number we have Therefore, from parts (a) and (b) we have Hence, the three cube roots of are 4. This time we have Therefore the four fourth roots of are 5. For we have Therefore, Thus, 