Let r and s be integers. Then ( 6^(r+s) x 12^(r-s) ) / ( 8^(r) x 9^(r+2s) ) is an integer when: (a) r+s <= 0, (b) s <= 0, (c) r <= 0, (d) r >= s.

First of all, an integer is a whole number, i.e. a not a fractional number, that can be positive, negative or zero. To find the range of values r and s for which the above fraction is an integer we can start by simplify the expression. By separating out the powers of 2 and 3 at both the numerator and denominator we thus obtain: (2x3)^r+s x (2²x3)^r-s / ( 2^3r x 3^2(r+2s) ) which then equals to 2^-s x 3^-4s. This is an integer when s <= 0. The right answer is (b). Note that r does not affect the final value of the expression. 

The notions to remember here are the exponent rules and properties: (1) Product rules: aⁿ x a^m = a^n+m and (a x b)ⁿ=aⁿ x bⁿ. (2) Quotient rules: aⁿ / a^m = a^n-m. (3) Power rules: (aⁿ)^m = a^nxm. (4) Negative exponent: a^-n = 1/(aⁿ).