There are three assumptions used when scientists measure ages with the radiometric dating process. 1.) The initial conditions of the fossil are known. In other words, the scientist assumes that neither the parent nor the daughter atoms have ever been altered outside of the decaying process since the formation of the fossil. 2.) The rate of decay has always been constant. This goes back to the idea of uniformitarianism, which is strongly adhered to by those who study historical science. The rate of decay could have been faster or slower in the past due to numerous factors. 3.) There has been no contamination of the elements. None of the atoms have leached out of the fossil, and none of the atoms have been added to the fossil. Helium, for example, is known to leach out of a specimen. Uranium and lead are leached by ground water.
These three assumptions are important to know when performing measurements, but it is unwise to make these assumptions because although the test results may produce long ages, how can you really know the test is valid? An interesting note: when specimens are tested and result in a young age, they are assumed to be contaminated and then discarded.
To help you better understand the three assumptions of radiometric dating, please see this example.