What represents the simplified expression of (3x^(-1) - 3y^(-1))/(y^(-2) - x^(-2))?

To simplify the given expression ((3x^{-1} - 3y^{-1})/(y^{-2} - x^{-2})), we can first manipulate both the numerator and the denominator.

Starting with the numerator, we can factor out (3):

[

3x^{-1} - 3y^{-1} = 3(x^{-1} - y^{-1})

]

Next, we can express (x^{-1}) and (y^{-1}) in terms of their common denominator, which is (xy):

[

x^{-1} - y^{-1} = \frac{y - x}{xy}

]

Substituting this back into the numerator gives:

[

3(x^{-1} - y^{-1}) = 3 \frac{y - x}{xy} = \frac{3(y - x)}{xy}

]

Now, let's move on to the denominator. We can recognize that (y^{-2} - x^{-2}) can be factored as a difference of squares:

[

y^{-2} - x^{-2} = (y^{-1} - x^{-1})(y^{-1} + x^{-1