Evaluate a nested radical-reciprocal sum: If m = 7 − 4√3, compute √m + 1/√m.

Introduction / Context:Numbers of the form a − 2√(ab) + b are perfect squares: (√a − √b)^2. Recognizing m in this way allows immediate simplification of √m and then of its reciprocal, producing an elegant integer sum.Given Data / Assumptions:

• m = 7 − 4√3.
• Compute S = √m + 1/√m.

Concept / Approach:Observe 7 − 4√3 = (2 − √3)^2 because (2 − √3)^2 = 4 + 3 − 4√3 = 7 − 4√3. Then √m = 2 − √3. Its reciprocal is found by rationalizing: 1/(2 − √3) = 2 + √3 (since (2 − √3)(2 + √3) = 1).

Step-by-Step Solution:

Verification / Alternative check:Compute m numerically: 7 − 4*1.732 ≈ 0.072; √m ≈ 0.268; 1/√m ≈ 3.732; sum ≈ 4.000.

Why Other Options Are Wrong:

• 3, 8, 9: Do not match the exact evaluation using the perfect-square recognition and rationalization.

Common Pitfalls:Missing the perfect-square pattern or making sign mistakes when rationalizing the denominator.

Final Answer: