There is a point between the Moon and the Earth where the gravitational attractions are equal and opposite. How much further is this point from the Earth than the Moon
Using the formula for gravitational attraction between the Moon/Earth and a test mass, m. We determine FE = (G*ME m)/(rE2 ), FM = (GMM*m )/ (rM2). Setting these two forces equal to each other and rearranging gives us the formula: rM2 = (MM/ME ) * rE2. Taking the square root of both sides, and noting the distances, r, should always be positive we get distance from the Earth is sqrt(MM/ME).
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