If the moon were twice as far from the earth as it is today how would the gravitational force it exerts on the earth compare with the force it exerts today?

The mutual gravitational force between two objects is inversely proportional to the square of the distance between their centers of mass. That goes for any two objects in the universe, regardless of their relative size or mass, the distance between them, how they're moving, or where in creation they're located. So if the distance between the earth and moon were doubled, the mutual force between them would change by a factor of ( 1 / 22 ) = 1/4 Each body would experience a force directed toward the other one, equal to one fourth of what the force is now.