Find all cube roots of \(8\) in the complex plane. Write your answers in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and justify your answer.
Solution
Write \(8=8e^{2\pi i k}\) for \(k\in\mathbb Z\). Its cube roots are \[ 2e^{2\pi i k/3},\qquad k=0,1,2. \] Thus the three roots are \[ 2, \qquad 2\left(-\frac12+\frac{\sqrt3}{2}i\right)=-1+\sqrt3\,i, \qquad 2\left(-\frac12-\frac{\sqrt3}{2}i\right)=-1-\sqrt3\,i. \] Cubing each gives \(8\), and a degree-\(3\) polynomial has at most three roots, so this list is complete.