What is the Relation between Power and the Rpm in an Electrical Motor....

You've left out torque...torque is the force acting on the motor shaft, work is that force over distance, and power is how fast that work is done...

https://www.engineeringtoolbox.com/electrical-motors-hp-torque-rpm-d_1503.html

"The third pump affinity law deals with hydraulic power. The power that a particular pump generates varies with the cube of the proportional speed change, and thus is most impacted by speed adjustments. The affinity law equation for power is:

P1/P2 = [N1/N2]³

Where N1 = Original speed, P1 = Original power, N2 = New speed, P2 = New power. Rearranging we get, P2 = P1 X[N2/N1]³"

This equation solves for how much extra power is needed to increase the rpm a certain amount...

https://www.linkedin.com/pulse/affinity-laws-pump-nikhilesh-mukherjee

It has got less to do with the motor and more to do with what the motor is turning. The power consumed by the motor is simply proportional to the cube of the tip speed of what its driving. The figure refers to a fan, which has this characteristic.

That equation doesn't apply in general. Need to know type of motor, if it's an ordinary induction motor straight on the mains the speed only varies about 3% (slip) between no load and full load, and increasing power goes with decreasing speed.

If the motor is inverter driven so variable speed P₁/P₂ = (N₁/N₂)³ applies when torque varies as N², as for a centrifugal pump or fan, but not otherwise.

That "fan speed law" was most often used when deciding on the required motor HP when a particular fan or pump was connected via a belt drive or chain drive where the sprockets or sheaves could be changed to get the desired flow.

--JMM