At what point between the Earth and the Moon will their gravitational fields cancel each other?

As said before, gravity of moon and earth will cancel each other at the five Lagrangian points. For further information you could read the article at Wikipedia:
kelseymh6 years ago
Proving that there are five Lagrangian points and determining their locations is left as a homework problem.
There are FIVE points in the earth moon system where gravity "cancels" - they are called the Lagrangian points.