Surd manipulation: Let x = √3 + √2. Evaluate (x + 1)/(x − 1) − (x − 1)/(x + 1).

Introduction / Context:
This problem tests algebraic manipulation of surds and rationalizing expressions involving sums and differences with square roots.

Given Data / Assumptions:
x = √3 + √2; compute (x + 1)/(x − 1) − (x − 1)/(x + 1).

Concept / Approach:
Use the identity A/B − B/A = (A^2 − B^2)/(AB). Set A = x + 1 and B = x − 1 to simplify quickly.

Step-by-Step Solution:
Let A = x + 1, B = x − 1.Then A/B − B/A = (A^2 − B^2)/(AB).A^2 − B^2 = [(x + 1)^2 − (x − 1)^2] = [x^2 + 2x + 1] − [x^2 − 2x + 1] = 4x.AB = (x + 1)(x − 1) = x^2 − 1.So the expression = 4x/(x^2 − 1).Compute x^2: (√3 + √2)^2 = 3 + 2 + 2√6 = 5 + 2√6.Thus x^2 − 1 = 4 + 2√6 = 2(2 + √6) and 4x = 4(√3 + √2) = 2·2(√3 + √2).Hence expression = [2·2(√3 + √2)] / [2(2 + √6)] = [2(√3 + √2)] / (2 + √6).Rationalize: multiply top and bottom by (2 − √6). Simplification yields √2.

Verification / Alternative check:
Substitute approximate values √3 ≈ 1.732, √2 ≈ 1.414 to confirm numerically ≈ 1.414.

Why Other Options Are Wrong:
√3 and √6 do not match the simplified exact value; “None of above” is incorrect because a specific root fits.

Common Pitfalls:
Expanding incorrectly or forgetting to use A/B − B/A identity, and errors while rationalizing.

Final Answer:
√2