Which property of exponents must be used first to solve this expression? (xy^2)1/3

I think you meant (xy^2)^(1/3), where 1/3 is the exponent of (xy^2).

Normally you must perform any operations showin inside parentheses first. But here xy^2 cannot be simplified. Your first step, then, is to apply the exponent (1/3) to (xy^2). Note that you're not "solving" this expression; rather, you're "evaluating it."

(xy^2)^(1/3) comes out to x^(1/3)*y^(2/3). This expression cannot be simplified further.