Final answer:
After three half-lives, 1/8 of the initial radioactive sample remains.
Step-by-step explanation:
When a radioactive sample decays, it decreases in amount by half after each half-life. After one half-life, half of the original sample remains, which would be 1/2 of the initial amount. After two half-lives, you're left with half of that amount, which is 1/4.
Therefore, after three half-lives, you again take half of the preceding amount (1/4), leaving you with 1/8 of the initial sample. Thus, the fraction of the radioactive sample that remains after it has decayed for three half-lives is 1/8.