Final answer:
The age of a rock after three half-lives would be the product of three multiplied by the length of one half-life. For instance, a rock with a radioactive isotope that has a half-life of 2 billion years would be 6 billion years old after three half-lives.
Step-by-step explanation:
If three half-lives have passed, a radioactive element in a sample would have decreased to 1/8 of its original amount. To determine the age of the rock, we multiply the number of half-lives by the length of one half-life. For example, if the half-life of a radioactive isotope is 2 billion years, after three half-lives (3 x 2 billion years), the rock would be 6 billion years old.
Depending on the element being considered, its half-life, and the context, the specific age may vary. For Rb-87 with a half-life of 4.7 x 10¹⁰ years, after three half-lives, only an eighth of the original element remains, indicating that the rock is very old, on the order of billions of years.
In cases where multiple radioactive isotopes are present, such as 238 U, 226 Ra, 222 Rn, and 210 Po, we can still find them in old rocks due to their chain of decay. Even though the half-lives of Rn and Po are much shorter, they are continually replenished by the decay of their longer-lived precursors, such as 238 U.