A man standing on the Earth can exert the same force with his legs as when he is standing on the moon. We know that the mass of the man is the same on the Earth and the moon.
We also know that F = ma is true on both the Earth and the morn. Will the man be able to
jump higher on the moon than the Earth? Why or why not?

No

Explanation:

From the analogy of the problem we are made to know that "a man standing on the earth can exert the same force with his legs as when he is standing on the moon".

 This force he is exerting is due to his weight. If he can have the same weight on the earth and moon, therefore:

      weight  = mass x acceleration due gravity

His mass and acceleration due to gravity on both terrestrial bodies are the same.

So, his jump height will be the same on earth and on the moon.

In summary, we have been shown that his mass and the acceleration due to gravity on both planets are the same, therefore, his weight will also be the same. His jump height will also be same.

mistakes are ok