Calculate remaining nuclei after decay.
Radioactive decay is a constant-probability process, so the surviving population halves over every half-life regardless of how many nuclei are present. Raising one half to the power of the number of elapsed half-lives gives the surviving fraction directly, without needing exponentials. Because the decay is exponential rather than linear, a sample is never fully gone — after ten half-lives just under a thousandth of the original remains.
Exponential decay by half-lives
half-lives = elapsed time / half-life; N = N0 x 0.5 ^ half-lives; decayed = N0 - N
Educational calculation. Radiation safety decisions require qualified assessment and compliance with applicable regulations.
The usual working rule is ten half-lives, leaving about 0.1 percent of the original activity. For technetium-99m at six hours that is two and a half days; for iodine-131 at eight days it is nearly three months.
It gives the number of decay events. If the daughter is stable, that equals the daughter nuclei formed. If the daughter is itself radioactive, its population depends on both decay constants and needs a decay-chain treatment.
They are mathematically identical — 0.5 to the power of t/T equals e to the power of minus lambda t. The half-life form is easier to sanity-check, since three half-lives visibly means one eighth remains.