Algebraic identity with radicals (minimal repair for clarity): Let x = √a + 1 and y = √a − 1 with a > 0. Evaluate the expression (x^4 + y^4 − 2x^2 y^2) / [ (√a)(√a) ].

Introduction / Context:Expressions like x^4 + y^4 − 2x^2 y^2 simplify dramatically using the identity x^4 + y^4 − 2x^2 y^2 = (x^2 − y^2)^2. With x and y written in terms of √a, the task reduces to computing a simple difference and then dividing by (√a)(√a) = a.

Given Data / Assumptions:

• x = √a + 1, y = √a − 1, with a > 0.
• Compute (x^4 + y^4 − 2x^2 y^2) / a.

Concept / Approach:First compute x^2 − y^2, because the target expression equals (x^2 − y^2)^2 / a. Squaring binomials with radicals requires care but is routine. Finally, divide by a to obtain a constant.

Step-by-Step Solution:

Verification / Alternative check:Recognize symmetry: replacing 1 by −1 in y changes only the middle term of (√a ± 1)^2, making the difference double that middle term; squaring then removes the radical.

Why Other Options Are Wrong:5, 10, and 20 stem from arithmetic slips like squaring 4√a incorrectly (e.g., writing 4a or 8a) or forgetting to divide by a.

Common Pitfalls:Miscomputing (√a ± 1)^2 or neglecting that (√a)(√a) = a leads to wrong constants.

Final Answer:16