Is power a cumulative quantity? Is the power needed to do a particular work cumulative? Like, the power needed to do work W for one second is P, is the power needed to do the same work for 2, 3, 4... seconds equal to 2P, 3P, 4P...?
 A: Power is work divided by time. Or the rate at which work is done. So the average power required to do the same amount of work in twice the time would be $\frac{P}{2}$
I exactly don't know what cumulative quantity means. But I feel a cumulative quantity is something which adds up over time or space (like mass, distance, work). Power is not one such quantity. If work is analogous to the distance traveled, then power is analogous to the instantaneous speed.
I believe you might be thinking of work as an instantaneous quantity. 1J of work for 1s and 1J of work for 2s, are the same amount of work. Work is the "total" quantity here. (Work x Time) isn't a quantity that we calculate, or have a name for
A: You can hoover a room 1 second, 10 seconds, or one hour with a vacuum cleaner of 2000(W). You don't need to add power. If the room becomes twice as big, the energy needed becomes twice that big.
You would need less time if you hoovered with higher power. Power is the energy used per second, Joule/second, J/s. Action is the energy times seconds, Js.
So the energy used is the power times the seconds using that power, and the action made/done/acted/performed is the energy used times the using time.
So you can do have/make/do/act lot of action with high power in a lot of time. But don't put too much power in the vacuum cleaner. Chaos might increase.
