Powers and Roots of Complex Numbers

Example:

Calculate and plot (((1 + i))^n) for n = 1, …, 10.

p  = Table[(((1 + I))^n), {n, 1, 10}]

In polar form: 1 + i  =  (Sqrt[2]) (e^(i π/4)), so (((1 + 1))^n) =  
        (2^(n/2)) (e^(i n π/4)). Thus, the angle bewteen successive points is 45 degrees, and the distance of the points from the origin increases exponentially by powers of (Sqrt[2]).

ListPlot[Table[{Re[p[[n]]], Im[p[[n]]]}, {n, 1, 10}], 
      PlotStyle -> PointSize[0.02], 
      AspectRatio -> Automatic]

Do[ListPlot[{{Re[p[[n]]], Im[p[[n]]]}}, 
        PlotStyle -> PointSize[0.02], 
        PlotRange -> {{-5, 30}, {-10, 35}}, 
        AspectRatio -> Automatic], {n, 1, 10}]

Example

Find the 8th roots of 3 -  4  i.

s  =  Solve[(z^8) == 3 - 4 I]

ListPlot[Table[{Re[z], Im[z]} /. s[[n]], {n, 1, 8}], 
      PlotStyle -> PointSize[0.02], 
      AspectRatio -> Automatic]

In polar form: 3  -  4  i  =  5 (e^(i α))where α  =  arctan(-4/
              3) =  -0.9272952180016122`. So the principle 8th root is \(TraditionalForm\`\((3 - 4  i)\)\^\(1/8\) = \ \(5\^\(1/8\)\ e\^\(i\ α/8\)\  = \(\(1.2228445449938519`\)\(\ \)\(e\^\(\(-0.11591190225020152`\) i\)\)\(\ \)\)\)\). The other roots are 
\(TraditionalForm\`1.2228445449938519`\ e\^\((\(-0.11591190225020152`\)\  + \ \ 2  π\ k/8)\)\) for k  =  1,  2,  …,  7.

(5^(1/8)) // N
ArcTan[-4/3]/8  // N

z  =  
        1.22284 (E^((-0.115912  +  k π/4) I));
ListPlot[Table[{Re[z], Im[z]}, {k, 0, 7}], 
      PlotStyle -> PointSize[0.02], 
      AspectRatio -> Automatic]