We know that a^2b^3=108. We can factor 108 into its prime factors: 108 = 2^2 x 3^3. We can then rewrite a^2b^3 as (a^2)(b^3) = (2^2)(3^3).
Since a and b are positive integers, we can see that a^2 must be one of the factors of 2^2, and b^3 must be one of the factors of 3^3. The only pairs of factors that multiply to 108 and satisfy these conditions are (a^2, b^3) = (2^2, 3^3) and (a^2, b^3) = (1, 108).
The first pair gives us a=2 and b=3, while the second pair gives us a=1 and b=108. Since a and b are positive integers, we can discard the second solution, so we have a=2 and b=3.
Finally, we can substitute these values into the expression 2a + 3b to get:
2a + 3b = 2(2) + 3(3) = 4 + 9 = 13.
Therefore, the value of 2a + 3b is 13.