jmc

The prime factorization of $360=2^3\cdot 3^2\cdot 5$. If a number is a perfect square, all of the exponents in its prime factorization must be even. Thus we need to multiply by a 2 and a 5, for a product of 10, which is the minimum possible value of x. Similarly, y can be found by making all the exponents divisible by 3, so the minimum possible value of $y$ is $3\cdot 5^2=75$. Thus, our answer is $x+y=10+75=\boxed{85}$.