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

The correct option is 1/(2√2).

To find the fraction of the original radioactive activity that remains after a certain amount of time, we can use the radioactive decay formula based on the number of half-lives. The formula is:

$\frac{N}{N_0}={\left(\frac{1}{2}\right)}^n$

Where:

- $N$ is the final amount of activity remaining.
- $N_0$ is the initial amount of activity.
- $n$ is the number of half-lives that have elapsed.

First, we calculate the number of half-lives ($n$) by dividing the total time elapsed ($t$) by the half-life ($T_{1/2}$):

$n=\frac{t}{T_{1/2}}=\frac{150}{100}=\frac{3}{2}$

Now, substitute $n=\frac{3}{2}$ into the decay formula to find the fraction remaining:

$\frac{N}{N_0}={\left(\frac{1}{2}\right)}^{\frac{3}{2}}$

We can simplify this exponential expression step-by-step:

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

A fractional exponent of 3/2 is equivalent to cubing the number and taking the square root:

$\frac{1}{2^{3/2}}=\frac{1}{\sqrt{2^3}}$

Since $2^3=8$, we have:

$\frac{1}{\sqrt{8}}$

Finally, we can simplify the radical by factoring out a perfect square ($8=4·2$), taking the square root of 4 out of the radical:

$\frac{1}{\sqrt{8}}=\frac{1}{2\sqrt{2}}$

Thus, the fraction of the original activity that will remain is $\frac{1}{2\sqrt{2}}$.