400g decayed to 50g in 24 days. Find the half-life t_(1)/(2).

$N_0 = 400 \text{ g}$
$N(t) = 50 \text{ g}$
$t = 24 \text{ days}$
We need to find $T$.

Divide both sides by 400.

Since $8 = 2^3$, then $\frac{1}{8} = \left(\frac{1}{2}\right)^3$.

The half-life is $\boxed{8 \text{ days}}$.

$T = 4 \text{ years}$
$N_0 = 256 \text{ g}$
$N(t) = 16 \text{ g}$
We need to find $t$.

Divide both sides by 256.

Since $16 = 2^4$, then $\frac{1}{16} = \left(\frac{1}{2}\right)^4$.

The time elapsed is $\boxed{16 \text{ years}}$.

$T = 3 \text{ hours}$
$N(t) = 512 \text{ g}$
$t = 12 \text{ hours}$
We need to find $N_0$.

Multiply both sides by 16.

The original mass was $\boxed{8192 \text{ g}}$.