The following problem consists of two equations:
1. 2^(X) + 2^(Y) = 2^(X+Y)
2. 3^(X) * 3^(Y) = 3^(X*Y)
We observe that these equations hold true only when X = Y.
For the first equation, given 2^(X) + 2^(Y) = 2^(X+Y), if X equals Y, the equation becomes:
2^(X) + 2^(X) = 2^(2X),
which simplifies to:
2 * 2^(X) = 2^(X+X),
which is then simplified as:
2 * 2^(X) = 2^(2X),
Indeed, 2 * 2^(X) does equal 2^(2X) for any X. Therefore, the equality holds true for any and all integer values of X when X equals Y.
For the second equation, given 3^(X) * 3^(Y) = 3^(X*Y), if X equals Y, the equation becomes:
3^(X) * 3^(X) = 3^(X*X),
which is essentially:
(3^(X))^2 = 3^(X^2),
The left side and the right side of the equation are equal for any X. Therefore, the equation holds true for any integer values of X when X equals Y.
Thus, the solution to the given problem is the set of all pairs (X, Y) with X = Y. This set includes an infinite number of pairs of integers, because it includes (1,1), (2,2), (3,3), ..., and so on for all integers.
Therefore, there are infinitely many pairs of integers (X, Y) that satisfy either of the two given equations.