The half-life of a radioactive element is $1600$ years. The fraction of the sample that remains undecayed after $6400$ years will be:

$A$ 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:

If a radioactive material remains $25 \%$ after $16$ days,then its half-life will be ......... days.

Radioactive nuclei $P$ and $Q$ disintegrate into $R$ with half-lives $1$ month and $2$ months respectively. At time $t=0$,the number of nuclei of each $P$ and $Q$ is $x$. At the time when the rates of disintegration of $P$ and $Q$ are equal,the number of nuclei of $R$ is ........ $x$.

$A$ sample has a half-life of $10^{33}$ years. If the initial number of nuclei of the sample is $26 \times 10^{24}$,then the number of nuclei decayed in $1$ year is ........... $\times 10^{-7}$.

The half-life period of a radioactive sample is $T$. If a fraction $x$ disintegrates in time $t$,what fraction will decay in time $\frac{t}{2}$?