In an algebraic expression involving cubes, if x^3 - y^3 = 81 and the difference x - y is equal to 3, what is the value of the sum of squares x^2 + y^2?

Introduction / Context:
This algebra question checks your understanding of identities involving cubes and squares, and how to connect different expressions like x^3 - y^3, x - y, x^2 + xy + y^2, and x^2 + y^2. Problems of this kind are very common in aptitude exams because they require both conceptual knowledge of standard identities and careful algebraic manipulation without necessarily solving for x and y individually.

Given Data / Assumptions:

• x^3 - y^3 = 81.
• x - y = 3.
• We need the value of x^2 + y^2.
• x and y are real numbers for which the given relations are valid.

Concept / Approach:
The key identity is x^3 - y^3 = (x - y)(x^2 + xy + y^2). We can use the given values of x^3 - y^3 and x - y to find x^2 + xy + y^2. Then we relate x^2 + xy + y^2 to x^2 + y^2 and xy using two different identities: the cube identity and the square of a difference identity (x - y)^2 = x^2 + y^2 - 2xy. Solving these small equations will give us xy, and then x^2 + y^2 directly.

Step-by-Step Solution:
From x^3 - y^3 = (x - y)(x^2 + xy + y^2). Substitute the given values: 81 = 3 * (x^2 + xy + y^2). So x^2 + xy + y^2 = 81 / 3 = 27. We also know (x - y)^2 = x^2 + y^2 - 2xy. Given x - y = 3, so (x - y)^2 = 9. Thus x^2 + y^2 - 2xy = 9. Call this equation (1). From x^2 + xy + y^2 = 27, we get x^2 + y^2 = 27 - xy. Call this (2). Substitute x^2 + y^2 from (2) into (1): (27 - xy) - 2xy = 9. This simplifies to 27 - 3xy = 9. So 3xy = 27 - 9 = 18, giving xy = 6. Now use (2): x^2 + y^2 = 27 - xy = 27 - 6 = 21.

Verification / Alternative check:
We can quickly check consistency. If x - y = 3 and xy = 6, then x and y are roots of t^2 - (x + y)t + xy = 0, but we do not need explicit values. Instead, compute x^2 + xy + y^2 = (x^2 + y^2) + xy = 21 + 6 = 27, which matches our earlier result. Substituting back into (x - y)(x^2 + xy + y^2) gives 3 * 27 = 81, which is exactly the given x^3 - y^3. This confirms that x^2 + y^2 = 21 is correct.

Why Other Options Are Wrong:
18 and 24 come from incorrect manipulation of identities or arithmetic errors when isolating xy. 27 is the value of x^2 + xy + y^2, not x^2 + y^2. 36 is too large and would not satisfy both identities at the same time. Only 21 is consistent with all given conditions.

Common Pitfalls:
Typical mistakes include confusing x^2 + xy + y^2 with x^2 + y^2, forgetting the factor x - y in the cube identity, or misusing (x - y)^2. Some students try to solve for x and y directly, which is unnecessary and can introduce extra errors. Working symbolically with identities is faster and safer in exam conditions.

Final Answer:
The required value of x^2 + y^2 is 21.