What is the initial recoil for a resting submarine with a mass that is 2.5 greater than 10 kg to the sixth power when it fires a 250-kg torpedo with an initial speed of 100 mps?

There is no recoil. Torpedoes are launched by one of two methods: either by water impulse (primary) or by motor startup within the tube.

Impulse is achieved when the water in the in the torpedo tube contacts one side of a piston ram that is located in a separate chamber below. An air flask dumps compressed air into the other side of the piston, forcing the water and torpedo out of the tube simultaneously, where the motor starts. Since the tube is flooded prior to impulse, it's equalized to the sea pressure of the depth the submarine is at. The force of the impulse isn't even close to the force necessary to affect an opposing force on a submarine.

If the impulse system isn't working, the tube can be flooded and the motor can be started in the tube and the torpedo can exit on its own.

All modern torpedoes have their own active/passive location-and-tracking sonars; once they've got a dead lock, you're history, unless you're in a boat or ship that can outrun a torpedo that is traveling toward you at 55 knots.