The half-life of a radioactive nuclide is 100 hours. The fraction of original activity that will remain after 150 hours would be :

The decay of a radioactive substance follows the exponential law:

$N(t)=N\_0\backslash left(\backslash frac\{1\}\{2\}\backslash right)^\{t/T\_\{1/2\}\}$

where:

$N(t)$ is the activity after time $t$,
$N\_0$ is the initial activity,
$T\_\{1/2\}$ is the half‑life.

Given a half‑life $T\_\{1/2\}=100\backslash \; \backslash text\{hours\}$ and a time interval $t=150\backslash \; \backslash text\{hours\}$, the exponent becomes:

$\backslash frac\{t\}\{T\_\{1/2\}\}=\backslash frac\{150\}\{100\}=1.5$

Therefore the remaining fraction is:

$\backslash left(\backslash frac\{1\}\{2\}\backslash right)^\{1.5\}$

Rewrite the exponent as a mixed fraction:

$1.5=\backslash frac\{3\}\{2\}$

So

$\backslash left(\backslash frac\{1\}\{2\}\backslash right)^\{\backslash frac\{3\}\{2\}\}=\backslash left(\backslash frac\{1\}\{2\}\backslash right)^\{1\}\backslash times\backslash left(\backslash frac\{1\}\{2\}\backslash right)^\{\backslash frac\{1\}\{2\}\}$

The first factor is simply $\backslash frac\{1\}\{2\}$. The second factor is the square root of $\backslash frac\{1\}\{2\}$, which equals $\backslash frac\{1\}\{\backslash sqrt\{2\}\}$. Multiplying the two factors gives:

$\backslash frac\{1\}\{2\}\backslash times\backslash frac\{1\}\{\backslash sqrt\{2\}\}=\backslash frac\{1\}\{2\backslash sqrt\{2\}\}$

Hence, after 150 hours the fraction of the original activity that remains is:

$1/(2\backslash sqrt\{2\})$