If one uses the root command instead of 1/3 as the exponent to guarantee that you get the principle root, you get an immediate simplification for
to 1. The function form
f2 := a -> (a-2)/(a-root(8,3));
works perfectly at 2, but gives 1 for any variable input.
With the first example, if one uses simplify on the denominator for a=1/3 one gets 0. Shouldn't a check for a zero denominator always be done?
q := (6*((1/3)*a-1/9))/(36*a-116+12*sqrt(12*a^3-3*a^2-54*a+93))^(1/3);
Finally, a question: Is the basic problem in Maple's evaluation of this expressions that it does not check for a 0 denominator?