An electric fan with effective area of cross-section ‘A’ accelerates air of density ‘$\mathrm{\rho }$ ’ to a speed ‘v’, the power needed for this process is

$\mathrm{Average} \mathrm{power}=\frac{\mathrm{Kinetic} \mathrm{energy} \mathrm{transferred} }{\mathrm{time}}$

Let us assume in 't' seconds 'm' mass of air is accelerated

Kinetic energy transferred = $\frac{1}{2}{\mathrm{mv}}^2$; here $\mathrm{m}=\mathrm{\rho }\left(\mathrm{Avt}\right)$

So the Average power $\mathrm{p}=\frac{1}{2}{\mathrm{\rho AV}}^3$