Powers of Complex Numbers: Geometric Visualization of Rotation and Scaling

① This example demonstrates how raising a complex number to a power (e.g., z^n) affects its modulus and argument in the complex plane, showing that the modulus scales exponentially and the argument rotates by a multiple. ② Core mathematical concepts include polar representation of complex numbers, De Moivre's theorem, geometric interpretation of complex multiplication, and rotation on the unit circle. ③ Construction highlights: using GeoGebra commands such as Intersect, Circle, Rotate, Angle to build the unit circle and rotation angles; ToComplex and ToPolar for coordinate conversions; Vector and Mirror for displaying vectors and symmetries. Key objects include N, A, c, C, Nroot, α_1, u, v, α, NrootPower, w, z, β, with slider n controlling the power, Zpower showing the result, and Zshow toggling visibility. ④ Suitable for high school to early college students to grasp the geometric meaning of complex powers, and useful for math experiments or inquiry-based learning.