Evaluate a mixed surd expression (clarified power placement): If a = 2 + √3, find the exact value of a^2 + a^(−2).

Introduction / Context:
Numbers of the form 2 ± √3 are algebraic conjugates. Expressions like a^2 + a^(−2) often simplify to integers by using the symmetric identity involving (a + 1/a). This approach avoids messy surd expansions and is a standard trick in surds-and-indices problems.

Given Data / Assumptions:

• a = 2 + √3 (so 1/a = 2 − √3).
• Compute a^2 + a^(−2).

Concept / Approach:
Let S = a + 1/a. Then S^2 = a^2 + 2 + a^(−2). Hence a^2 + a^(−2) = S^2 − 2. For a = 2 + √3, the reciprocal is 2 − √3, so S is an integer, making the result an integer as well.

Step-by-Step Solution:

Verification / Alternative check:
Direct expansion: a^2 = (2 + √3)^2 = 7 + 4√3; a^(−2) = (2 − √3)^2 = 7 − 4√3; sum = 14. This confirms the identity-based approach.

Why Other Options Are Wrong:

• 12, 16, 18: These are near misses that arise from squaring and forgetting to subtract 2, or arithmetic slips.

Common Pitfalls:
Misreading the power as a^2 + a − 2 (which would not be an integer). The clarified expression a^2 + a^(−2) is standard and yields a clean integer result.

Final Answer: