The expression $\frac{x + y}{x - y} \div \frac{(x + y)^2}{(x^2 - y^2)}$ is equal to

### Concept & Formula This problem tests the simplification of rational expressions involving algebraic identities and fraction division. The key rule to remember is how to divide fractions (multiply by the reciprocal) and the difference of squares factorization. $$A^2 - B^2 = (A - B)(A + B)$$ ### Step-by-Step Solution Given expression: $\frac{x + y}{x - y} \div \frac{(x + y)^2}{x^2 - y^2}$ **Step 1: Convert division to multiplication** To divide by a fraction, we multiply by its reciprocal (flip the second fraction). $= \frac{x + y}{x - y} \times \frac{x^2 - y^2}{(x + y)^2}$ **Step 2: Factor the expressions** Use the difference of squares identity on the numerator of the second fraction: $x^2 - y^2 = (x - y)(x + y)$. Expand the denominator for clarity: $(x + y)^2 = (x + y)(x + y)$. Substitute these factored forms back into the expression: $= \frac{x + y}{x - y} \times \frac{(x - y)(x + y)}{(x + y)(x + y)}$ **Step 3: Cancel common terms** Now, look for identical binomials in the numerators and denominators to cancel out. - The $(x - y)$ in the denominator cancels the $(x - y)$ in the numerator. - The two $(x + y)$ terms in the numerator completely cancel the two $(x + y)$ terms in the denominator. $= \frac{1}{1} \times \frac{1}{1}$ $= 1$ ### Exam Strategy & Shortcut **Value Putting (Zero Assumption):** To solve this in 5 seconds, pick the easiest possible numbers that don't result in division by zero. Let $x = 2$ and $y = 1$. First part: $\frac{2+1}{2-1} = \frac{3}{1} = 3$. Second part: $\frac{(2+1)^2}{2^2 - 1^2} = \frac{3^2}{4-1} = \frac{9}{3} = 3$. Expression becomes: $3 \div 3 = 1$. This completely avoids algebra and is foolproof. ### Common Pitfall A very common mistake is confusing the $\div$ symbol with a $+$ sign due to poor printing quality on exam papers. Always double-check the operator before starting long algebraic simplifications. Additionally, forgetting to flip the second fraction when converting from division to multiplication is a frequent arithmetic error. ### Final Answer Therefore, the correct answer is **1**.