If the earth suddenly shrinks to (1)/(64)th of its original volume...

$L =$ constant; $L = I\omega$

From the conservation of angular momentum,

$\dfrac{2}{5}M{R}^{2}{\omega}_{1} = \dfrac{2}{5}M{\left(\dfrac{R}{4}\right)}^{2}{\omega}_{2} ⇒ M{R}^{2}{\omega}_{1} = \dfrac{M{R}^{2}{\omega}_{2}}{16} ⇒ \dfrac{{\omega}_{1}}{{\omega}_{2}} = \dfrac{1}{16}$

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

$\dfrac{{T}_{2}}{{T}_{1}} = \dfrac{1}{16}$

$⇒ \dfrac{{T}_{1}}{{T}_{2}} = \dfrac{16}{1}$

Using ${T}_{1} = 24h,$$x = 16$ by comparing with the given time period in the question.