If a^4 + 1 = (a^2 / b^2) * (4b^2 – b^4 – 1) for non–zero integers a and b, then what is the value of a^4 + b^4?

Introduction / Context:
This question involves an algebraic equation that links the powers of two variables a and b. The aim is to use the given relationship to find a specific symmetric expression a^4 + b^4. Such problems test your ability to manipulate powers and fractions and to interpret seemingly complicated expressions in a structured way.

Given Data / Assumptions:

• a and b are non zero integers.
• The equation a^4 + 1 = (a^2 / b^2) * (4b^2 - b^4 - 1) holds.
• We need to find the value of a^4 + b^4.

Concept / Approach:
Since a and b are restricted to integers, we look for integer solutions that satisfy the given equation. It is often effective to test small integer values that keep the algebra manageable. By checking small positive and negative values, we can find all integer pairs (a, b) that satisfy the equation and then compute a^4 + b^4 for those pairs. If every valid integer pair produces the same value of a^4 + b^4, that value is the required answer.

Step-by-Step Solution:
Start with the equation a^4 + 1 = (a^2 / b^2) * (4b^2 - b^4 - 1). Multiply both sides by b^2 to remove the denominator: (a^4 + 1) b^2 = a^2 (4b^2 - b^4 - 1). Because a and b are integers, test small integer values such as a = ±1 and b = ±1. Take a = 1 and b = 1. Then left side is 1^4 + 1 = 2. Right side is (1^2 / 1^2) * (4*1^2 - 1^4 - 1) = 1 * (4 - 1 - 1) = 2. So the equation is satisfied. Take a = -1 and b = 1. Left side is (-1)^4 + 1 = 2. Right side is ((-1)^2 / 1^2) * (4*1^2 - 1^4 - 1) = 1 * 2 = 2. The equation is again satisfied. Similarly, a = 1, b = -1 and a = -1, b = -1 also satisfy the equation, since a^2 and b^2 remain 1. For each of these valid integer pairs, compute a^4 + b^4. Since a^4 = 1 and b^4 = 1 for a = ±1 and b = ±1, we get a^4 + b^4 = 1 + 1 = 2. No other small integer values of a and b satisfy the equation, so among non zero integers, the only possibilities give a^4 + b^4 = 2.

Verification / Alternative check:
We can check that trying a = 0 or b = 0 is not allowed, because the equation contains the term a^2 / b^2, which would be undefined if b is zero. Testing other small integers such as a = 2, b = ±1 or a = ±2, b = ±2 quickly shows that the given relation fails. Hence the only integer solutions are based on ±1 and ±1, and they all lead to the same value a^4 + b^4 = 2.

Why Other Options Are Wrong:
16, 32, 64: These correspond to cases where a and b would have absolute values greater than 1, for example 2, 3 or 4, but such pairs do not satisfy the given equation when checked.
8: This might result from assuming a^4 and b^4 are different small powers such as 1 and 7, but that combination does not satisfy the original relation.
Only the value 2 is consistent with all valid integer solutions of the equation.

Common Pitfalls:
A frequent difficulty is to attempt full symbolic manipulation instead of using the integer condition to guide trial values. This can become very messy and time consuming. Another issue is ignoring the restriction that a and b are non zero and accidentally trying b = 0, which makes the expression undefined. A systematic test of small integer values is usually the fastest and most reliable method in such contest style questions.

Final Answer:
The value of a^4 + b^4 is 2.