The problem given in the image is to simplify the following expression:

$\frac{x^2 - y^2}{(x - y)^2} \times \frac{1}{x + y}$

Step-by-step simplification:

1. Recognize that $x^2 - y^2$ is a difference of squares: $x^2 - y^2 = (x - y)(x + y)$

   Now substitute this into the original expression: $\frac{(x - y)(x + y)}{(x - y)^2} \times \frac{1}{x + y}$

2. Simplify the first fraction: $\frac{(x - y)(x + y)}{(x - y)^2} = \frac{x + y}{x - y}$ (since $(x - y)$ cancels out from the numerator and denominator).

3. Multiply the resulting expression by $\frac{1}{x + y}$: $\frac{x + y}{x - y} \times \frac{1}{x + y} = \frac{1}{x - y}$

Thus, the simplified expression is:

$\boxed{\frac{1}{x - y}}$

Would you like further details or have any questions?

Here are 5 related questions for practice:

1. Simplify $\frac{x^2 - 9}{(x - 3)(x + 3)} \times \frac{1}{x + 3}$.
2. Simplify $\frac{a^2 - b^2}{(a - b)^2} \times \frac{1}{a + b}$.
3. Simplify $\frac{4x^2 - 25}{(2x - 5)^2} \times \frac{1}{2x + 5}$.
4. Simplify $\frac{z^2 - 16}{(z - 4)(z + 4)} \times \frac{1}{z + 4}$.
5. Multiply and simplify $\frac{x^3 - y^3}{(x - y)^3} \times \frac{1}{x + y}$.

Tip:

When simplifying rational expressions, always look for common factors that can cancel out, especially for differences of squares, sums of cubes, or factorizable polynomials.