Radical symmetry trick: If x = 2 + √3, evaluate (√x + 1/√x).

Introduction / Context:Expressions of the form √x + 1/√x can often be evaluated without explicitly finding √x by using the identity (√x + 1/√x)^2 = x + 1/x + 2. The key is to compute x + 1/x first, which is straightforward for numbers of the form a + √b.

Given Data / Assumptions:

• x = 2 + √3.
• We need the exact value of √x + 1/√x.
• x is positive, so √x is real and positive.

Concept / Approach:Compute 1/x by rationalizing the denominator: 1/(2 + √3) = (2 − √3)/(4 − 3) = 2 − √3. Then add x + 1/x to leverage the square identity to find √x + 1/√x.

Step-by-Step Solution:

Verification / Alternative check:Let t = √x. Then t^2 = x and (t + 1/t)^2 = t^2 + 1/t^2 + 2 = x + 1/x + 2. Plug in the computed value to confirm t + 1/t = √6.

Why Other Options Are Wrong:√3 and 2 are too small; 2√6 and 6 are too large and come from squaring without taking the square root properly.

Common Pitfalls:Forgetting to add the extra +2 when using (√x + 1/√x)^2, or rationalizing 1/x incorrectly.

Final Answer:√6