The prime factorization of 792 is 3^3*3^2*11^1. For an integer to be divisible by 792, its prime factorization must include 2, 3, and 11 with exponents of at least 3, 2 and 1 respectively.
For an integer to be a perfect square, the exponents in its prime factorization must be even. The smallest integer that meets this condition as well as the condition for divisibility by 792 is 2^4*3^2*11^2. This differs from 792 by having one more factor of 2 and one more factor of 11. Thus, it is 22*792 and the smallest integer Joelle could have multiplied 792 by to get a perfect square is 22.