A) What are the possible radiations emitted by radioactive nuclei? b) If a radioactive sample has a half-life of 24 days and an initial quantity of 6.4 x 108 atoms, how many atoms will remain after 96

Given:
Half-life ($T_{1/2}$) = 24 days
Total time ($t$) = 96 days

$$n = \frac{t}{T_{1/2}}$$ $$n = \frac{96 \text{ days}}{24 \text{ days}}$$ $$n = 4$$

Given:
Initial number of atoms ($N_0$) = $6.4 \times 10^{8}$
Number of half-lives ($n$) = 4

$$N = N_0 \left(\frac{1}{2}\right)^n$$ $$N = (6.4 \times 10^{8}) \left(\frac{1}{2}\right)^4$$ $$N = (6.4 \times 10^{8}) \left(\frac{1}{16}\right)$$ $$N = \frac{6.4 \times 10^{8}}{16}$$ $$N = 0.4 \times 10^{8}$$ $$N = 4.0 \times 10^{7} \text{ nuclei}$$

Number of decayed nuclei = Initial number of nuclei - Number of remaining nuclei

$$N_{decayed} = N_0 - N$$ $$N_{decayed} = (6.4 \times 10^{8}) - (4.0 \times 10^{7})$$ $$N_{decayed} = (64 \times 10^{7}) - (4.0 \times 10^{7})$$ $$N_{decayed} = (64 - 4) \times 10^{7}$$ $$N_{decayed} = 60 \times 10^{7}$$ $$N_{decayed} = 6.0 \times 10^{8} \text{ nuclei}$$

The number of nuclei that would have decayed after 96 days is \boxed{\text{6.0 \times 10^8 nuclei}}.