$A$ radioactive element has a half-life period of $800$ years. After $6400$ years,what fraction of the initial amount will remain?

$A$ radioactive source has a half-life of $6 \,h$. $A$ freshly prepared sample of the same exhibits radioactivity $32$ times the permissible safe value. The minimum time after which it would be possible to work safely with the source is (in $\,h$)

For a substance,the average life for $\alpha$-emission is $1620$ years and for $\beta$-emission is $405$ years. After how much time will $1/4$ of the material remain after both $\alpha$ and $\beta$ emissions?

The ratio of the number of active nuclei of two different radioactive samples is $2:3$. Their half-lives are $1 \ h$ and $2 \ h$ respectively. The ratio of the number of active nuclei after $6 \ h$ will be:

$A$ radioactive isotope has a half-life of $T$ years. How long will it take the activity to reduce to $(a)$ $3.125\%$ $(b)$ $1\%$ of its original value?

In a radioactive material,the fraction of active material remaining after time $t$ is $\frac{9}{16}$. The fraction that was remaining after $\frac{t}{2}$ is: