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Finding real roots of negative numbers (for example, $\sqrt\[3\]{-8}$)

I am trying to make Mathematica plot the cube roots of $27i$ and graph them, so that I can include them in my $\LaTeX$ed homework. How can I do this?

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marked as duplicate by rm -rf♦, Verbeia♦, Oleksandr R., J. M.♦ Sep 22 '12 at 2:43

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1 Answer

up vote 5 down vote accepted

From @belisarius

pts = ({Re@#, Im@#} & /@ (x /. Solve[x^3 == 27 I]))

Styling:

Show[ContourPlot[Abs[(x + I y)^3 - 27 I], {x, -4, 4}, {y, -4, 4}, Contours -> 15],
 Graphics[{{Style[Text[#, #], 17] & /@ #}, {Opacity[.5], Orange, 
      Thickness[.01], Arrow[{{0, 0}, #}] & /@ #}, {Red, 
      PointSize[.02], Point@#}}, Axes -> True, Frame -> True, 
    PlotRangePadding -> 1, AspectRatio -> Automatic] &@pts]

enter image description here

Show[ContourPlot[Arg[(x + I y)^3 - 27 I], {x, -4, 4}, {y, -4, 4}, 
  Contours -> 15, ColorFunction -> "Rainbow"],
 Graphics[{{Style[Text[#, #], 17] & /@ #}, {Opacity[.5], Orange, 
      Thickness[.01], Arrow[{{0, 0}, #}] & /@ #}, {Red, 
      PointSize[.02], Point@#}}, Axes -> True, Frame -> True, 
    PlotRangePadding -> 1, AspectRatio -> Automatic] &@pts]

enter image description here

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