If $12^2 \cdot 18^3 = 2^x \cdot 3^y$, find $x+y$.
I notice that $12$ and $18$ are both multiples of $2$ and $3$, so I can use the prime factorization of these numbers to rewrite them as powers of $2$ and $3$. Using the exponent rules, I get $12^2 \cdot 18^3 = (2^2 \cdot 3^1)^2 \cdot (2^1 \cdot 3^2)^3 = 2^4 \cdot 3^2 \cdot 2^3 \cdot 3^6 = 2^7 \cdot 3^8$. Now I can compare the exponents of $2$ and $3$ on both sides of the equation and set them equal. I get $x = 7$ and $y = 8$. Therefore, $x+y = 15$. 