A half-life is the time a quantity takes to fall to half of whatever it currently is — and crucially, it does not matter what it currently is. $N = N_0\left(\tfrac{1}{2}\right)^{t/t_{1/2}}$: read aloud, N equals N-nought, times one half, to the t over t-half. The symbol $t_{1/2}$ is said “t-half”. That exponent is doing one simple job — counting how many halvings fit into the elapsed time.

So the working move is to count FIRST. Three half-lives is $\tfrac{1}{2}\times\tfrac{1}{2}\times\tfrac{1}{2} = \tfrac{1}{8}$ left, and you knew that before you touched a calculator. One half, a quarter, an eighth, a sixteenth: the powers of two are the most useful eight numbers in this course, and the magnitude stage in this lesson exists to make you reach for them first.

The continuous version writes the same physics with a decay constant: $\lambda = \dfrac{\ln 2}{t_{1/2}}$, read lambda equals ell-en two over t-half, where $\lambda$ is the Greek letter lambda. It is a rate, so it wears per-time units, and a LONG half-life gives a SMALL lambda — a check worth running every single time, because it catches the upside-down answer without any arithmetic.