I think that Jim's argument is quite correct. It looks as if we need a dramatically increased WorkingPrecision here:

sol = FindRoot[9^(3 x) - 12 x^x + 45 x^3 - 66 == 0, {x, 726.509, 726.513}, MaxIterations -> 1000000, WorkingPrecision -> 10000]

This is only the first part of the solution! With that we get:

9^(3 x) - 12 x^x + 45 x^3 - 66 /. sol
(*0.*10^-7917*)

which is a numerical zero. The function is just quite steep at that point.

N[D[9^(3 x) - 12 x^x + 45 x^3 - 66, x] /. sol, 10]
(*-6.325431641*10^2079*)

Best wishes, Marco

The last terms are of no consequence to the overall shape of the curve. What we have is the contention between the 9^(3x) term and the -12 x^x term for dominance. Below 727, the first term dominates, but then it is then overtaken by the second. I knew that the x^x term was going to win in the long term, but I didn't think it would take that long or be so 'sudden' in terms of the numbers.

Ok, Wolfram Alpha gave the correct root, 0.630231, but also gave the bogus root of about 726. Just plotting the equation shows that the second root is an error.

However, I used Mathematica, and used FindRoot[] rather than Solve[] or NSolve[], due to the complexity of the expression. It came up with the single root, 0.630231 (approximately), and no other in the reals. The function is only defined for x>0.

I clicked on the cloud icon in W|A and got the code that it tried for Solve, Reduce and NReduce, none of which gave the second result. (These functions also did not give the first, correct result.)

Back in Mathematica, changing the method for FindRoot[] also failed to find a root.

SO, I believe that there is no bug in Mathematica itself. I already sent feedback to Wolfram Alpha about this.

The first course of action with problems with Wolfram|Alpha should be to provide feedback on the W|A page. They usually respond fairly quickly.

Also, before the Moderation team responds, it is generally bad form on this forum to announce a bug the way you did, especially with no context.

Yes indeed there would:

FindRoot[9^(3 x) - 12 x^x + 45 x^3 - 66,  {x, 727}  ]

{x -> 726.511}

@max ron: Maybe you should consider the following:

f[x_] := 9^(3 x) - 12 x^x + 45 x^3 - 66
Sign[f[Rationalize[726.5105]]]
(* 1 *)
Sign[f[Rationalize[726.5110]]]
(* -1 *)

Between a positive value and a negative value there's got to be a zero.