This was very important before computers in problems where you had to do something else after computing an answer.
One simple example is the following: When you calculate the angle between two vectors, often you get a fraction containing roots. In order to recognize the angle, whenever when possible, it is good to have a standard form for these fractions [side note, I saw often students not being able to find the angle $\theta$ so that $\cos(\theta)=\frac{1}{\sqrt{2}}$]. The simplest way to define a standard form is by making the denominator or numerator integer.
If you wonder why the denominator is the choice, it is the natural choice: As I said often you need to make computations with fractions. What is easier to add:
$$\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{6}+\sqrt{3}} \, \mbox{ or }\, \frac{\sqrt{3}}{3}+\frac{\sqrt{6}-\sqrt{3}}{3} \,?$$
Note that bringing fractions to the same denominator is usually easier if the denominator is an integer. And keep in mind that in many problems you start with quantities which need to be replaced by fractions in standard form [for example in trigonometry, problems are set in terms of $\cos(\theta)$ where $\theta$ is some angle].
But at the end of the day, it is just a convention. And while you think that $\frac{1}{\sqrt{2}}$ looks simpler, and you are right, the key with conventions is that they need to be consistent for the cases where you need recognition. The one which looks simpler is often relative...
Best Answer
It turns out that division is the same as multiplying something by its reciprocal. In this case, $\frac{1}{\frac{1}{\sqrt{2}}}=1\cdot\frac{\sqrt{2}}{1}=\sqrt{2}$, as desired. In general, to divide by a fraction, just multiply by the reciprocal.