Final answer:
To reduce the activity of a radioactive isotope to 1/8 of its initial value, it takes 3 half-lives, or 3T, as the activity decreases by half with each half-life.
Step-by-step explanation:
Understanding the Half-life of Radioactive Isotopes
The half-life (T) of a radioactive isotope is the time it takes for half of the original amount of the substance to decay. To find out how long it will take for the activity to reduce to 1/8 of its initial value, we must consider multiple half-lives. After one half-life, the substance's activity would be reduced to 1/2 its initial value; after two half-lives, to 1/4; and after three half-lives, to 1/8. Therefore, it would take 3 half-lives (3T) for a radioactive isotope's activity to reduce to 1/8 of its initial value.