Let's simplify the expression:

$\frac{\left(\frac{x}{y^2} - \frac{y}{x^2}\right)}{\left(\frac{1}{y^2} - \frac{1}{x^2}\right)}$

Step 1: Simplify the numerator

The numerator is:

$\frac{x}{y^2} - \frac{y}{x^2}$

Find the common denominator:

$\frac{x}{y^2} - \frac{y}{x^2} = \frac{x \cdot x^2 - y \cdot y^2}{y^2 \cdot x^2} = \frac{x^3 - y^3}{x^2 y^2}$

So the numerator simplifies to:

$\frac{x^3 - y^3}{x^2 y^2}$

Step 2: Simplify the denominator

The denominator is:

$\frac{1}{y^2} - \frac{1}{x^2}$

Find the common denominator:

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

So the denominator simplifies to:

$\frac{x^2 - y^2}{x^2 y^2}$

Step 3: Simplify the overall expression

Now, the entire expression is:

$\frac{\frac{x^3 - y^3}{x^2 y^2}}{\frac{x^2 - y^2}{x^2 y^2}} = \frac{x^3 - y^3}{x^2 - y^2}$

Step 4: Factor the expression

• The numerator $x^3 - y^3$ is a difference of cubes, which can be factored as:

$x^3 - y^3 = (x - y)(x^2 + xy + y^2)$

• The denominator $x^2 - y^2$ is a difference of squares, which can be factored as:

$x^2 - y^2 = (x - y)(x + y)$

Step 5: Cancel common factors

Now the expression becomes:

$\frac{(x - y)(x^2 + xy + y^2)}{(x - y)(x + y)}$

Since $x - y$ appears in both the numerator and denominator, we can cancel it (assuming $x \neq y$):

$\frac{x^2 + xy + y^2}{x + y}$

Thus, the simplified form of the expression is:

$\boxed{\frac{x^2 + xy + y^2}{x + y}}$

Let me know if you want further details or clarifications!

Here are 5 related questions to further your understanding:

1. How would the simplification change if $x = y$?
2. Can you apply this method to simplify other rational expressions?
3. How do you factor the sum of cubes instead of the difference of cubes?
4. What happens if you try to simplify the expression without factoring?
5. How would you simplify similar expressions involving higher powers of $x$ and $y$?

Tip: Always check if the numerator and denominator can be factored, as it often leads to significant simplifications.