If average of m numbers is n^2 and average of n numbers is m^2 then average of m+n numbers is

The sum of the m numbers divided by m (which is the average) equals n^2. Then, the sum of the m numbers equals mn^2.

The sum of the n numbers divided by n (which is the average) equals m^2. Then, the sum of the n numbers equals nm^2.

The average of m+n numbers which is the sum of the m numbers plus the sum of the n numbers divided by (m+n) equals (mn^2+nm^2) / (m+n). This is mn (n+m) / (m+n). Then the factor (m+n) can be ruled out and the result is mn.