simplify("(x^2)^0.5") should be "abs(x)"

[paulftw]

While there are genuine use cases when returning "x" would be better (that's what most humans would expect), strictly speaking it is incorrect.

To reproduce in the mathnotepad:

derivative("(x^2)^0.5", "x").evaluate({x:-3})
    1

derivative("(x^2)^0.5", "x", {simplify:false}).evaluate({x:-3})
    -1

Here I evaluate the same derivative at the same point x=-3 but get two different results.
Same problem for x^4 and I assume all other even powers.

[josdejong]

Thanks that is a very good and valid point.

Anyone able to improve simplify? Help would be welcome.

[gwhitney]

Well, the relation (x^2)^0.5 = abs(x) is not an identity for complex numbers, e.g, (i^2)^0.5 = i but abs(i) = 1. But I completely agree that (x^2)^0.5 = x is not an identity in general either, for example with negative numbers as Paul points out. I think the only tenable approach is to further restrict use of such an identity, or eliminate it altogether from simplify, rather than replacing it with (x^2)^0.5 = abs(x); or possibly having both with different restrictions (e.g., if x is known to be a real number, then I guess (x^2)^0.5 = abs(x) is OK.