Question 639861
Your answer is correct. Here is how I get there.
The first operation we need to do is get rid of the negative exponents.
This is done by replacing them with their inverse (also called the reciprocal). 
For example,
a^(-1) = 1/a
Likewise
b^(-1) = 1/b
and 
a^(-3) = 1/a^3
and
b^(-3) = 1/b^3
Make these sustitutions in your expression yields
(1) (1/a - 1/b)/(1/a^3 - 1/b^3)
Now let's get rid of the fractions.
Multiply (1) by (a^3*b^3)/(a^3*b^3), which changes nothing because we are multiplying by one, and obtain
(2) (a^2*b^2)((b - a)/(b^3 - a^3))
The denominator of (2) factors into
(3) (b - a)*(b^2 + ab + a^2)
You can FOIL (3) to show that it is equal to the denominator of (2).
Substituting (3) into (2) which simplifies to
(4) (a^2*b^2)/(b^2 + ab + a^2)
The answer (4) is the same as your "belief", but now you know the rest of the story.
We can also go from (1) to (2) by cross multiplication to subtract the fractions in the numerator and denominator of (1).
The numerator of (1) is
(5) (1/a - 1/b) = (b-a)/(ab)
and the denominator of (1) is
(6) (1/a^3 - 1/b^3) = (b^3 - a^3)/(a^3*b^3)
Taking the ratio of (5)/(6) yields (2).