Answer
$0.2\frac{rev}{s}$.
Work Step by Step
The angular momentum is conserved. It the rotational inertia multiplied by the angular velocity.
$$I_i\omega_i=I_f\omega_f$$
$$\omega_f =\omega_i\frac{I_i}{I_f}$$
$$=\omega_i\frac{\frac{2}{5}MR_i^2}{\frac{2}{5}MR_f^2}$$
$$=\omega_i\frac{R_i^2}{R_f^2}$$
$$=(1rev/month)\frac{(6\times10^6m)^2}{(8000m)^2}$$
$$=(5.625\times10^5 \frac{rev}{month})\frac{month}{30d}\frac{d}{86400s}\approx0.2\frac{rev}{s}$$
