If $a = 11$ and $b = 9$, then the value of $$ \left( \frac{a^2 + b^2 + ab}{a^3 - b^3} \right) $$ is

### Concept & Formula This problem can be rapidly solved by recognizing standard algebraic identities rather than calculating large numerical powers. The key formula required is the difference of cubes: $$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$ ### Step-by-Step Solution * **Given:** $a = 11$ and $b = 9$. We need to evaluate the expression: $\frac{a^2 + b^2 + ab}{a^3 - b^3}$. * **Expansion:** Substitute the denominator $a^3 - b^3$ with its expanded identity form: $$\frac{a^2 + ab + b^2}{(a - b)(a^2 + ab + b^2)}$$ * **Simplification:** Notice that the numerator $(a^2 + ab + b^2)$ perfectly cancels out with the corresponding term in the denominator. $$\frac{1}{a - b}$$ * **Calculation:** Substitute the given values of $a$ and $b$ into the simplified fraction: $$\frac{1}{11 - 9} = \frac{1}{2}$$ ### Exam Strategy & Shortcut **Algebraic Cancellation:** Never plug numbers directly into expressions involving squares and cubes if the structure looks like a standard identity. Always expand the higher-degree term (in this case, the denominator) to look for cancellation opportunities. This reduces a multi-minute calculation into a 10-second mental math problem. ### Common Pitfall Students often waste time calculating $11^2 = 121$, $11^3 = 1331$, $9^3 = 729$, etc. This not only consumes precious time but also vastly increases the probability of basic arithmetic errors under exam pressure. Always look for algebraic simplification first! ### Final Answer Therefore, the correct answer is **1/2**.