Insect lands on a tree with sap seeping out. Sap is really sticky, and occasionally the insect becomes stuck there and dies. Sap encases the insect and eventually solidifies into amber with insect trapped inside.
Radiometric dating is the process where scientists observe isotopes of a particular atom in a substance. If scientists know what the original distribution of the isotope is compared to when the substance first formed and if they know the half-life of the atom, they can mathematically calculate how old the thing is. Carbon dating is a form of this using a carbon isotope that builds up to a certain level when an organic thing is alive but then decays once that thing dies. By measuring the ratio of carbon isotopes, scientists can calculate about when the thing died. Carbon dating is reliable up to around 50,000 years. For older things, other types of radiometric dating are more often used.
Other types of dating often involve examining the geology around a sample. If a sample is found at a particular layer in the crust, its age is likely to be the same as other things found in the same layer. So if a particular sample is difficult to test for whatever reason, using geology can help place it with other samples that are more reliably testable.
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