Answer:
Newton gives us force
F=-G*M*m/R^2
Divide by the mass of the object to get acceleration
a=F/m = GM/R^2
You are solving the distance where the force from the moon equals the force from the Earth -
a= G*(mass of earth)/R^2 - G*(mass of moon)/(384400000-R)^2=0
Or in other words
G*(mass of earth)/R^2=G*(mass of moon)/(384400000-R)^2
Divide by G*(mass of the moon) and multiply by R^2 to get terms on both sides
G*(mass of earth)/(G*(mass of moon))=R^2/(384400000-R)^2
Since the mass of the moon and earth are fixed, the left side is a number
81.3 = R^2/(384400000-R)^2 → R^2/(384400000-R)^2–81.3 = 0
simplify
R/(384400000-R) = 9.01665
R=346024000 = 346,024 km.
Step-by-step explanation: