Evaluate a nested radical sum with a specific parameter: Given a = √3 / 2, compute √(1 + a) + √(1 − a).

Introduction / Context:Radical expressions with symmetric terms √(1 + a) and √(1 − a) often simplify neatly for special values of a. Here, a = √3/2 is chosen so that trigonometric or algebraic identities lead to a simple exact value.Given Data / Assumptions:

• a = √3/2.
• Compute S = √(1 + a) + √(1 − a).

Concept / Approach:We can square the sum S to eliminate radicals: S^2 = (1 + a) + (1 − a) + 2√((1 + a)(1 − a)) = 2 + 2√(1 − a^2). Then compute a^2 and simplify the inner square root.Step-by-Step Solution:

Verification / Alternative check:Approximate numerically: a ≈ 0.8660; √(1 + a) ≈ √1.8660 ≈ 1.3660; √(1 − a) ≈ √0.1340 ≈ 0.3660; sum ≈ 1.7320 = √3.

Why Other Options Are Wrong:

• (2 ± √3), √3/2: Do not match the exact sum derived from squaring and simplifying.

Common Pitfalls:Forgetting to add the cross term 2√((1 + a)(1 − a)) when squaring or mishandling the square root of a fraction.

Final Answer: