This is known as rationalizing the denominator (RTD). $ $ As the name suggests, it simplifies by transforming an irrational divisor into a rational divisor. As explained here, this is a prototypical instance of the method of simpler multiples. This can lead to all sorts of simplifications. Below are a couple prototypical examples.
In this prior question is an example where RTD transforms a limit of indeterminate form into a simple determinate limit by way of cancelling an apparent singularity at $\, x = a,\, $viz.
$$ \frac{x^2-a\sqrt{ax}}{\sqrt{ax}-a} = \frac{x^2-a\sqrt{ax}}{\sqrt{ax}-a} \ \frac{\sqrt{ax}+a}{\sqrt{ax}+a} = \frac{ax\,(x\!-\!a)+\sqrt{ax}\ (x^2\!-\!a^2) }{a\,(x-a) } = x+(x\!+\!a)\sqrt{\frac{x}{a}}$$
Here's a number-theoretic example showing how RTD reduces divisibility of algebraic integers to rational integers. Consider the Gaussian integers $\ \mathbb I = \{ m + n\ i\ : \ m,n\in \mathbb Z \},\,$ where $\,i^2 = -1.\,$ As usual we define divisibilty by $\ a\mid b\,\ {\rm in}\,\ \mathbb I \!\iff\! b/a \in \mathbb I,\,$ for $\,b\neq 0.\,$ Suppose we wish to know if $\ 2\!+\!3\, i\mid 91\ \,{\rm in}\,\ \mathbb I,\,$ i.e. is $\ w = 91/(2\!+\!3\, i)\in \mathbb I\ ?\ $ $\,\mathbb I\,$ happens to have a division algorithm which we could apply. But it is simpler to RTD: $\, w = 91\ (2\!-\!3\, i)/(2^2\!+\!3^2) = 7 (2\!-\!3\, i)\, $ so, indeed, $\, w\in \mathbb I$.
More generally we can often reduce problems about algebraic numbers to problems about rational numbers by taking norms, traces, etc. In fact this is how Kronecker constructed his divisor theory for algebraic integers, see e.g. Harold Edwards: Divisor Theory.