This is an enormous branch of geochemistry called Geochronology.
It is an accurate way to date specific geologic events.
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The reason that I trust the accuracy of the age that we have determined for the earth (~4.56 billion years) is that we have been able to obtain a very similar result using many different isotopic systems.
Most estimates of the age of the earth come from dating meteorites that have fallen to Earth (because we think that they formed in our solar nebula very close to the time that the earth formed).
Answer 2: Yes, radiometric dating is a very accurate way to date the Earth.
We know it is accurate because radiometric dating is based on the radioactive decay of unstable isotopes.
This is why crystals are good for radiometric dating: the atoms in a crystal are extremely efficiently packed, and it's very difficult to get anything into a crystal such as a contaminant by any means short of destroying the crystal and re-growing it anew.
The oldest crystals on Earth that were formed on Earth are zircon crystals, and are approximately 4.1 billion years old.
The ratio of the parent to daughter then can be used to back-calculate the age of that rock. The reason we know that radiometric dating works so well is because we can use several different isotope systems (for example, Uranium-Lead, Lutetium-Halfnium, Potassium-Argon) on the same rock, and they all come up with the same age.
This gives geologists great confidence that the method correctly determines when that rock formed.
We call the original, unstable isotope (Uranium) the "parent", and the product of decay (Lead) the "daughter".