Surd manipulation and rationalization If x = √3 + √2, find the exact value of x + 1/x.

Introduction / Context:
Questions of this type test comfort with surds and rationalizing denominators. The trick is to find 1/x by multiplying numerator and denominator by the conjugate.

Given Data / Assumptions:

• x = √3 + √2.
• We need x + 1/x.

Concept / Approach:
Use the conjugate of x, namely √3 − √2, to compute the reciprocal exactly. Then add the expressions and simplify.

Step-by-Step Solution:
1/x = 1/(√3 + √2) = (√3 − √2)/[(√3)^2 − (√2)^2] = (√3 − √2)/(3 − 2) = √3 − √2.Therefore, x + 1/x = (√3 + √2) + (√3 − √2) = 2√3.

Verification / Alternative check:
Let √3 ≈ 1.732, √2 ≈ 1.414. Then x ≈ 3.146; 1/x ≈ 0.318; sum ≈ 3.464, which equals 2√3 ≈ 3.464.

Why Other Options Are Wrong:
2 and 3 ignore surd terms; 2√2 and √6 come from adding or multiplying the roots incorrectly.

Common Pitfalls:
Using √3 + √2 as its own reciprocal or forgetting to divide by the difference of squares when rationalizing.

Final Answer:
2√3