drag force relationship with cube of speed?
You seem to be confusing force (thrust or drag) with energy (under which concept work and power comes). Don't. You'll be spared much grief.
Work done is force times distance.
$$ W = Fs $$
Power is work done divided by time
$$ P = W/t $$
When you double velocity, force quadruples. As does work.
$$ W_\text{new} = 4Fs = 4W $$
Time halves as well, with the following effect on power.
$$ P_\text{new} = 4W/0.5t = 8P $$