How can one accurately determine the exact age of rock with radioactive decay?

Answer

Radioisotope dating is a method of determining the age of (usually organic) material by looking at the ratio of stable and unstable isotopes.

For example, carbon dating uses the isotope carbon-14, which has a radioactive decay with a half-life of 5736 years. (So ​​every 5736 years only half of the original activity remains)
Carbon-14 (C-14 or ¹⁴C) is an isotope of carbon formed in our atmosphere from nitrogen nuclei. This happens through nuclear reactions as a result of the cosmic rays to which the earth is constantly exposed. In our environment there is therefore a certain fixed concentration of carbon 14, which occurs in addition to the stable isotopes carbon 12 and carbon 13. In organic material this is continuously absorbed and released (by e.g. CO₂). Once the organism dies, there is no more absorption and only radioactive decay remains. By measuring the activity of ¹⁴C one can find out how long ago the organism stopped exchanging ¹⁴C with its environment.
Nuclear tests (atomic bombs) in the years 1955-1980 ensured that the ¹⁴C in the environment (global ecosystem) is no longer a natural constant. For age determinations via ¹⁴C is usually compared to the ¹⁴C content before 1955.

For older fossils one can use other radioactive decay chains, for example those from uranium to lead or to thorium, or from potassium to argon.
This is usually based on the solidification of the rock and the stratigraphy is taken into account (for example by the principles of Charles Lyell), but the presence of fossils in these rocks is also considered. Furthermore, one can look at other benchmarks such as regional or global catastrophic events (major volcanic eruptions or meteorite impacts such as the meteorite impact that marks the boundary between the Cretaceous and Paleogene, and which deposited a thin layer of iridium-rich material worldwide).

For some radioactive (unstable) isotopes it is known in what amounts they occurred compared to stable isotopes when the rock was formed. By examining the ratio of these two types of isotopes in a rock, it is possible to calculate quite accurately when the rock must have formed. Not all rocks are suitable for radiometric dating.