Complex Analysis, Video #21 (Complex Arithmetic, Part 21).
Review polar form of positive integer powers. Review use of Manipulate and Locator on Mathematica to animate this process; use Graphics and Circle to create the unit circle and see how the unit circle forms a boundary between two types of behavior of powers of complex numbers (either going to infinity or going to zero). Start with a point on the unit circle (the point 1/2 + sqrt(3)/2*i) and see that the powers generate a periodic sequence of points. Use ComplexExpand to check it. Try it for 3/5 + (4/5)*i and see that it never comes back to itself. Find 1^(1/2) by thinking about polar form. Emphasize that this really represents all possible square roots (it's "multi-valued"). Get +1 and -1. Find 1^(1/3) by thinking about polar form as well. Get 1, e^(i*2pi/3) = -1/2 + i*sqrt(3)/2, and e^(i*4pi/3) = -1/2 - i*sqrt(3)/2. Check with ComplexExpand.