The decay constants of two radioactive elements are $15x$ and $3x$ respectively. Initially,they have the same number of nuclei. After a time of $\frac{1}{6x}$,the ratio of the number of their nuclei will be ........

The nuclear activity of a radioactive element becomes $\left(\frac{1}{8}\right)^{\text{th}}$ of its initial value in $30\, \text{years}$. The half-life of the radioactive element is $....\, \text{years}$.

If ${N_0}$ is the original mass of the substance of half-life period ${T_{1/2}} = 5 \text{ years}$,then the amount of substance left after $15 \text{ years}$ is

$A$ radioactive substance is being produced at a constant rate of $10 \text{ nuclei/s}$. The decay constant of the substance is $0.5 \text{ s}^{-1}$. After what time will the number of radioactive nuclei become $10$? Initially,there are no nuclei present. Assume the decay law holds for the sample.

For a substance,the fraction of its initial quantity $(N_0)$ which will disintegrate in its average lifetime is about $(e = 2.71)$.

$A$ sample contains $16\, g$ of a radioactive material, the half-life of which is $2\, days$. After $32\, days$, the amount of radioactive material left in the sample is: