Calculate the magnitudes of the gravitational forces exerted on the Moon by the Sun and by the Earth when the two forces are in direct competition, that is, when the Sun, Moon and Earth are aligned with the Moon between the Sun and the Earth. This alignment corresponds to a solar eclipse.) Does the orbit of the Moon ever actually curve away from the Sun, toward the Earth? (Please give your answer to three significant figures.)

I used G m1 m2/r^2 to get the forces, for the force between the Earth and the moon I did:
((6.67e-11)(5.97e24)(7.35e22))/(1.737e6 - 6.37e6)^2
and I got 1.36e24 N
To get the r value I substracted the radii... do I have to add them?
For the force between the Sun and the Moon, I used the distance from earth to the sun and substracted it from the distance between moon to earth.
((1.737e6 - 6.37e6)- (6.96e8 -6.37e6))=6.95e8 m

and then for the moon to sun force I computed:

((6.67e-11)(7.35e22)(1.9891e30))/(1.737e6 - 6.37e6)^2
and I got 1.40e34 N...

These answers are not right but I'm not sure what I'm doing wrong...

Why don't you figure Moon-Earth force, and moon-sun force?

Your distances: I have no idea what you did. You should be using center to center distances. What you did makes no sense to me.

To calculate the magnitudes of the gravitational forces exerted on the Moon by the Sun and the Earth when they are in direct competition, you need to use the correct formula and consider the distances accurately.

1. Calculating the force between the Earth and the Moon:
You correctly applied the formula for the gravitational force, which is F = (G * m1 * m2) / r^2, where G is the gravitational constant, m1 and m2 are the masses of the objects, and r is the distance between them.
However, when calculating the distance (r) between the Earth and the Moon, you subtracted their radii. That's not correct because you need to consider the actual centers of mass. The correct distance between the Earth and the Moon is the sum of their average distances from their centers: r = (1.737e6 + 3.84e8) m. Make sure to use the correct values.

2. Calculating the force between the Sun and the Moon:
When the Sun, Moon, and Earth are aligned with the Moon between the Sun and Earth (corresponding to a solar eclipse), the distance between the Moon and the Sun will be the sum of the distances from the Moon to the Earth and from the Earth to the Sun.
So, the correct distance (r) between the Sun and the Moon is the sum of their average distances from their centers: r = (1.737e6 + 1.496e11) m.

After correcting the distances, you can recalculate the forces between the Moon and the Earth, and between the Moon and the Sun using the formulas you provided:
Force between the Earth and the Moon = (6.67e-11 * 5.97e24 * 7.35e22) / (3.84e8)^2
Force between the Sun and the Moon = (6.67e-11 * 1.9891e30 * 7.35e22) / (1.496e11)^2

By evaluating these expressions correctly, you can obtain the magnitudes of the gravitational forces exerted on the Moon by the Sun and the Earth when they are in direct competition.