-( (a^2) / (b^3) )^3 (-a^2 b^2)^3
= -( a^6 / b^9) (-a^6 b^6)
= -a^6 * -a^6 * b^(6-9)
= a^12 * b^-3
Did I simplify this correctly? The difficult part is the -(a^6) and (-a^6).
Since the red negative sign doesn't show up in any parenthesis it is simply a constant factor out in front of everything. For the black negative sign think of it this way:
(-a^2 b^2)^3 = (-1)^3 * (a^2)^3 * (b^2)^3 = -(a^2)^3 * (b^2)^3
which is again outside of everything. So you got it right.