If x = a + 1/a and y = a − 1/a for a ≠ 0, what is the value of the expression x⁴ + y⁴ − 2x²y²?

Introduction / Context:
This algebra problem involves symmetric expressions and powers of two derived variables x and y. The given definitions of x and y in terms of a and 1/a suggest using identities for squares and differences to simplify the expression x⁴ + y⁴ − 2x²y² without expanding everything fully.

Given Data / Assumptions:

• x = a + 1/a.
• y = a − 1/a.
• a is nonzero.
• We want to find x⁴ + y⁴ − 2x²y².

Concept / Approach:
Main ideas:

• Observe that x⁴ + y⁴ − 2x²y² can be written as (x² − y²)².
• Compute x² and y² separately using binomial expansion.
• Find x² − y², then square it to obtain the final value.
• This avoids expanding the fourth powers directly.

Step-by-Step Solution:
Step 1: Note that x⁴ + y⁴ − 2x²y² = (x² − y²)², which is a standard identity for the square of a difference. Step 2: Compute x² = (a + 1/a)² = a² + 2 + 1/a². Step 3: Compute y² = (a − 1/a)² = a² − 2 + 1/a². Step 4: Find x² − y² = (a² + 2 + 1/a²) − (a² − 2 + 1/a²). Step 5: Simplify: x² − y² = a² + 2 + 1/a² − a² + 2 − 1/a² = 4. Step 6: Now compute (x² − y²)² = 4² = 16. Step 7: Therefore x⁴ + y⁴ − 2x²y² = 16.
Verification / Alternative check:
Step 1: Choose a specific nonzero value for a, for example a = 2. Step 2: Then x = 2 + 1/2 = 2.5 and y = 2 − 1/2 = 1.5. Step 3: Compute x² ≈ 6.25 and y² ≈ 2.25. Step 4: Compute x⁴ + y⁴ − 2x²y² numerically and you will find that the result is very close to 16, confirming the algebraic solution.
Why Other Options Are Wrong:
Option 4 is only the value of x² − y², not its square. Options 8 and 64 arise from incorrect squaring or misinterpreting intermediate results. Option 0 would require x² = y², which clearly is not the case because x and y are different when a is nonzero.
Common Pitfalls:
Learners sometimes try to expand x⁴ and y⁴ directly, which is lengthy and increases the chance of arithmetic mistakes. Another mistake is to misapply the identity for (x² − y²)² or to forget that x⁴ + y⁴ − 2x²y² has that special form. Students may also incorrectly expand (a ± 1/a)², missing the middle term or miscomputing 2(a)(1/a).
Final Answer:
The value of x⁴ + y⁴ − 2x²y² is 16.