e. \( \frac{x^{2}}{x^{2}-y^{2}}+\frac{x}{y-x} \)

Did you know that simplifying rational expressions like \( \frac{x^{2}}{x^{2}-y^{2}}+\frac{x}{y-x} \) can lead to fascinating connections in algebra? The expression can be analyzed by factoring the denominator, \( x^{2}-y^{2} \) as \( (x-y)(x+y) \), which helps us understand how different mathematical concepts come together, like the difference of squares! Additionally, when tackling expressions like this, it's easy to overlook the importance of finding a common denominator. A common mistake is jumping straight to adding without ensuring denominators match. By putting on your "math detective" hat and carefully working through each step—factoring, finding common denominators, and simplifying—you not only avoid errors but also gain a deeper mastery of the topic!