Nested surds identity (repaired for clarity): Evaluate √(5 + 2√6) − 1 / √(5 − 2√6).

Introduction / Context:This classic surds problem relies on recognizing conjugate forms generated by binomial squares. The original database text was ambiguous; applying the Recovery-First Policy, we clarify it as the well-known identity: √(5 + 2√6) − 1 / √(5 − 2√6).

Given Data / Assumptions:

• Expression: √(5 + 2√6) − 1/√(5 − 2√6)
• All radicals are principal square roots.

Concept / Approach:Notice (√3 + √2)^2 = 3 + 2 + 2√6 = 5 + 2√6, so √(5 + 2√6) = √3 + √2. Also (√3 − √2)^2 = 3 + 2 − 2√6 = 5 − 2√6, so √(5 − 2√6) = √3 − √2 (positive since √3 > √2). Then use the reciprocal of a conjugate: 1/(√3 − √2) = √3 + √2 after rationalizing.

Step-by-Step Solution:Let A = √(5 + 2√6) = √3 + √2Let B = √(5 − 2√6) = √3 − √2Compute 1/B = (√3 + √2) since B(√3 + √2) = (3 − 2) = 1Therefore expression = A − 1/B = (√3 + √2) − (√3 + √2) = 0

Verification / Alternative check:Approximate numerically: √(5 + 2√6) ≈ 3.146, √(5 − 2√6) ≈ 0.318; then 1/0.318 ≈ 3.146; the difference is ≈ 0.

Why Other Options Are Wrong:2√2, 2√3, and √5 − 1 are distractors that arise if you mishandle conjugates or skip rationalization.

Common Pitfalls:Forgetting that 1/(√3 − √2) equals √3 + √2 and not (√3 − √2)/(1). Rationalization is essential.

Final Answer:0