If the number 18225 can be prime factorized as 3^a × b², then what is the sum of a and b?

By the fundamental theorem of arithmetic, it is known that every composite number can be factorized as the product of primes in a unique way apart from the order in which the prime factors occur. 

The number 18225 can be prime factorized as:

18225 = 3 × 3 × 3 × 3 × 3 × 3 × 5 × 5 

= 3⁶ × 5²

Comparing 3⁶ × 5² with 3^a × b², we obtain a = 6 and b = 5

∴ a + b = 6 + 5 = 11

Thus, the sum of the numbers a and b is 11.