I only know a bit about this but from what I understand this is similar to how most modern mining works. Many minerals are mined in very low concentrations, which means they then need to be extracted out of the host material. Part of that process often involves multiple steps where the ore is soaked in various chemical solutions, which separate certain materials. It can be a very complicated process involving a lot of toxic chemicals.

The rare earth metals are kinda irritating to separate because they have very similar physical properties, but it obviously can be done. They frequently co-occur in ore deposits for exactly that reason.