What is the simplified form of (3x^(-1) - 3y^(-1))/(y^(-2) - x^(-2))?

To simplify the expression ((3x^{-1} - 3y^{-1})/(y^{-2} - x^{-2})), we begin by rewriting the powers of negative exponents in terms of their reciprocals.

1. The numerator (3x^{-1} - 3y^{-1}) can be rewritten as:

[

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

]

This shows that we can factor out (3) from the expression.

2. The denominator (y^{-2} - x^{-2}) can be rewritten as:

[

\frac{1}{y^2} - \frac{1}{x^2} = \frac{x^2 - y^2}{x^2y^2}.

]

The expression (x^2 - y^2) factors into ((x - y)(x + y)). Therefore, we have:

[

y^{-2} - x^{-2} = \frac{(x - y)(