One aspect of Mathematica probably deserves some emphasis in light of this question:
Sqrt[2]/4 autosimplifies back to 1/(2 Sqrt[2]).
Powers with the same base are combined; square roots are combined. The following sequence is like swimming upstream:
(-1 + Sqrt[3])/(2 Sqrt[2])
$\rightarrow$ Sqrt[2](-1 + Sqrt[3])/4
$\rightarrow$ (-Sqrt[2] + Sqrt[6])/4
If you don't stop the flow of evaluation, the second expression is swept downstream to the first automatically. The third expression is stable, if you can get there.
Luckily, there is a factor -1 + Sqrt[3], which can be multiplied term-by-term by Sqrt[2] to prevent Sqrt[2] from recombining with 4 (provided there is a way to do that before Sqrt[2]/4 autosimplifies). If you did not have such a factor, then there would be no way to keep the denominator rationalized.
That hopefully gives an idea of what one is up against in rationalizing the denominator. Other forms of simplification can also run against the flow of the autosimplification built into Mathematica.
Note that using Simplify[] or FullSimplify[] may or may not succeed in producing the desired form in each case. They try to produce an equivalent expression with a minimal number of leaves in its expression-tree. That's a distinct goal from rationalizing.
If formatting output is the main goal — by which I mean that you are not going to use the output as further input, but you are going to use it only for people to see and to read — then there are more ways. For instance, HoldForm[] or boxes.
