Let a be a positive real number and define x = a + 1/a and y = a - 1/a. What is the exact value of the expression x^4 + y^4 - 2x^2 y^2?

Introduction / Context:
This question tests your fluency with algebraic identities involving symmetric expressions in a and 1/a. Such patterns occur frequently in board exams and aptitude tests, where you are expected to simplify seemingly complicated expressions without expanding everything in terms of a directly.

Given Data / Assumptions:
- a is a positive real number, a > 0.
- x = a + 1/a.
- y = a - 1/a.
- We must find x^4 + y^4 - 2x^2 y^2.

Concept / Approach:
The key observation is that x^4 + y^4 - 2x^2 y^2 can be written as (x^2 - y^2)^2, which is a standard identity. Once we find x^2 and y^2 in terms of a and 1/a, we can compute x^2 - y^2 very easily, and then square it. This avoids a long expansion.

Step-by-Step Solution:
First compute x^2: x^2 = (a + 1/a)^2 = a^2 + 2 + 1/a^2. Next compute y^2: y^2 = (a - 1/a)^2 = a^2 - 2 + 1/a^2. Now find x^2 - y^2: x^2 - y^2 = (a^2 + 2 + 1/a^2) - (a^2 - 2 + 1/a^2) = 4. Use the identity x^4 + y^4 - 2x^2 y^2 = (x^2 - y^2)^2. Therefore, x^4 + y^4 - 2x^2 y^2 = 4^2 = 16.

Verification / Alternative check:
You can verify by choosing a convenient value for a, for example a = 2. Then x = 2 + 1/2 = 2.5 and y = 2 - 0.5 = 1.5. Numerically, x^4 is 2.5^4, y^4 is 1.5^4, and x^2 y^2 is (2.5^2)(1.5^2). Evaluating these and computing x^4 + y^4 - 2x^2 y^2 gives 16, confirming that the simplified result is correct and independent of the particular positive a chosen.

Why Other Options Are Wrong:
Values such as 20, 10, 8 or 5 can come from miscomputing x^2 or y^2, dropping the middle terms, or incorrectly using (x^2 + y^2)^2 instead of (x^2 - y^2)^2. None of these alternatives produce a constant value for all positive a, while 16 does.

Common Pitfalls:
A common error is to expand x^4 and y^4 directly in terms of a, which creates very long expressions and increases the chance of algebraic mistakes. Another mistake is to forget that (p - q)^2 = p^2 - 2pq + q^2 and instead assume a simpler pattern. Recognising the identity (x^2 - y^2)^2 saves both time and effort.

Final Answer:
The exact value of the expression is 16.