How do I rationalised and simplify surds?

Say I have a fraction with 2 surds, such as √10 / √6. To rationalise this, we need to get the dominator (bottom fraction) to be an integer. As √10 / √6 x 1 still gives √10 / √6, and we can write 1 as √6 / √6, we can do (√10 / √6) x (√6 / √6) = √(10 x 6) / 6 = √60 / 6.

This is now rationalised, as the demonator is no longer a surd, however it is not in its simplest form yet. To get this we can simplify √60 to give √(4 x 15) = √4 x √15 = 2√15. In the fraction this gives 2√15 / 6 which cancels down to √15 / 3.