If the earth suddenly shrinks to 1 64 th of its original volume with its mass remaining the same, the period…

$L=$ constant; $L=I\omega$

From the conservation of angular momentum,

$$\begin{gathered} \frac{2}{5}M{R}^2{\omega }_1=\frac{2}{5}M{\left(\frac{{\displaystyle R}}{4}\right)}^2{\omega }_2 \\ \Rightarrow M{R}^2{\omega }_1=\frac{{\displaystyle M{R}^2{\omega }_2}}{16} \\ \Rightarrow \frac{{\omega }_1}{{\omega }_2}=\frac{1}{16} \end{gathered}$$

Using $\omega =\frac{2\pi }{T}$ for each case along with the respective angular frequencies,

$\frac{T_2}{T_1}=\frac{1}{16}$

$\Rightarrow \frac{T_1}{T_2}=\frac{16}{1}$

Using $T_1=24 \mathrm{h},$$x=16$ by comparing with the given time period in the question.