We had a discussion on cubic roots three days ago in the thread "How can I use fractional exponents and find extraneous solutions?". The problem is how to interpret the cubic root of a negative number. There are two different schools of thought. The expression (-8)^(1/3) can be either -2 or a complex number, and each interpretation has its good reasons. Mathematica supports both, with two different commands: CubeRoot (or Surd) and Power. I don't know about Wolfram|Alpha. Try asking for the domain of CubeRoot[x^3-x]
