Plotting Odd Roots

The Difficulty

If you try to plot a third root or any odd root of a negative number you will get error messages.

Plot[(x^(1/3)),{x,-3,3}]

Roots

If we look at the roots of -1 we can see why this might be.

sol = Solve[(x^3)== -1,x]

solis= x/.sol

ptlis = Point[{Re[#],Im[#]}]&/@solis

Show[Graphics[{AbsolutePointSize[5],Hue[.95],ptlis}],AspectRatio -> Automatic,Axes -> True,AxesLabel -> { StyleForm["ℝ",FontWeight -> "Bold",FontSize -> 14], StyleForm["𝕀",FontSize -> 14,FontWeight -> "Bold"]},ImageSize -> 2.5 72,PlotRange -> All]

Notice that there are three roots as we would expect and the "first root" is a complex number in the first quadrant.  As a result of this and to make the results given by Mathematica as general as possible and to make the results consistent with even roots it was decided not to plot values of odd-roots for negative numbers.

A way around the problem

Here is a simple way around this problem. First we define a function.

OddRoot[x_,n_] := Which[x<0,-Abs[x^(1/n)],True,x^(1/n)]

And a plot

Plot[OddRoot[x,3],{x,-3,3}]