To find the value of ab, we can start by squaring both sides of the equation A + B = 4 to get:

(A + B)^2 = 4^2
A^2 + 2AB + B^2 = 16

Now, we are given that A^3 + B^3 = 40. We can also rewrite A^3 + B^3 as (A + B)(A^2 - AB + B^2) = 40, by using the formula for the sum of cubes.

Since we know that A + B = 4 and A^2 + B^2 = 16 - 2AB from the first equation above, we can substitute the values into the equation A^3 + B^3 = 40 to get:

4(16 - 2AB) = 40
64 - 8AB = 40
-8AB = -24
AB = 3

Therefore, the value of ab is 3.