The value of $\frac{(a + b)^2}{(a^2 - b^2)}$ is

### Concept & Formula This is a fundamental algebraic simplification problem based on factoring the **Difference of Squares**. $$A^2 - B^2 = (A - B)(A + B)$$ ### Step-by-Step Solution Given expression: $\frac{(a + b)^2}{a^2 - b^2}$ **Step 1: Expand and Factor** Rewrite the numerator by expanding the square into its two identical binomial factors: $(a + b)^2 = (a + b)(a + b)$ Apply the difference of squares identity to factor the denominator: $a^2 - b^2 = (a - b)(a + b)$ **Step 2: Substitute and Cancel** Place the expanded forms back into the fraction: $= \frac{(a + b)(a + b)}{(a - b)(a + b)}$ You can see that the binomial $(a + b)$ exists in both the numerator and the denominator. Cancel out one instance of $(a + b)$ from the top and the bottom (assuming $a + b \neq 0$): $= \frac{a + b}{a - b}$ ### Exam Strategy & Shortcut **Direct Recognition:** This is a basic identity relationship. You should ideally recognize instantly that $a^2 - b^2$ contains one $(a + b)$ factor which will cancel one of the $(a + b)$ factors on top, leaving exactly one $(a + b)$ on top and the $(a - b)$ on the bottom. This should take less than 3 seconds to answer without any calculation. ### Common Pitfall A common error among beginners is mistakenly expanding the numerator as $a^2 + b^2$, and then being unable to cancel anything with the denominator. Remember that $(a + b)^2 \neq a^2 + b^2$. ### Final Answer Therefore, the correct answer is **$\frac{a + b}{a - b}$**.