What is happening here? [Wolfram]

Question

When I input:

$2^{a/b} = (2^a)^{1/b}$

Into Wolfram Alpha, it produces a very strange looking "implicit plot":

I'm entirely sure what this means or what it's even trying to plot and how. I suspect it may be a bug, but I thought I'd ask in case anyone has an explanation.

Answer 1

WolframAlpha output is ultimately generated by Mathematica. That particular plot is generated by a Mathematica command that looks like so:

ContourPlot[2^(a/b) == (2^a)^b^(-1), {a, -1.3, 1.3}, {b, -1.3, 1.3}]

While, $2^{a/b} = (2^a)^{1/b}$ is true for a lot of real $a$ and $b$, Mathematica evaluates this numerically and gets some small errors. You can see this using a command like so:

Table[2^(a/b) - (2^a)^b^(-1), {a, 0.1, 0.5, 0.2}, {b, 0.1, 0.5, 0.2}]
(* Out:
  {{6.66134*10^-16, 2.22045*10^-16, 2.22045*10^-16}, 
   {0., 0., 0.}, 
   {-2.13163*10^-14, -4.44089*10^-16, -4.44089*10^-16}}
*)

Of interest here is the fact that there are positive, negative and zero values. Thus, the ContourPlot command generates the crazy looking graph. This can be fixed in Mathematica using the WorkingPrecision option.