Each half-life halves the remaining count, so three half-lives leave one eighth: 64 nuclei drop to 8 while 56 have decayed. The factor 64 / 8 = 8 and the decayed total 64 - 8 = 56 are exact for the stated isotope count budget; real decay measurements have branching ratios, detector efficiency, background counts, shielding, timing uncertainty, sample purity, and finite counting statistics.

Three exact half-lives leave one eighth of the count. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Three half-lives leave one eighth

Three exact halvings turn 64 into 8.

64÷8=864\div8=8
Three half-life ledgerThe power of two is checked by the helper.N0=64halves=3left=8decayed=56

The decayed total stays pinned

The decayed total is 56, not a rounded fit.

64−8=5664-8=56
Three half-life complementOnly integer half-life steps are in scope.N0=64halves=3left=8decayed=56