Answer
The energy that must be supplied to the centrifuge is $573~J$
Work Step by Step
We can find the rotational kinetic energy of the centrifuge:
$KE_{rot} = \frac{1}{2}I~\omega^2$
$KE_{rot} = \frac{1}{2}(6.5\times 10^{-3}~kg~m^2)~(420~rad/s)^2$
$KE_{rot} = 573~J$
Since the change in rotational kinetic energy is $573~J$, the energy that must be supplied to the centrifuge is $573~J$.
