How does a radiometric time scale measure the age of the Earth?
It doesn’t. I have not heard of a “radiometric time scale”, and I think what you meant to ask is something like: “How can radiometric techniques be used to date the age of the Earth?” Radioactive isotopes of various elements have established half-lives. This is based on the Physics principle of the Weak Internuclear Force. There are two basic methods that have been developed for dating rocks. One uses the ratio of parent isotopes to daughter isotopes while another uses ratios of daughter products having different parent isotopes. I am not going to go into any of the details, since these procedures are taught in Graduate-level Geology classes, and the terminology used is too complicated for someone without some knowledge of Physics. If you want more information type in