In aptitude (algebraic identities and surds), if x + y = sqrt(3) and x - y = sqrt(2), find the exact value of the expression 8 * x * y * (x^2 + y^2) using algebraic identities without approximating the square roots.

Introduction / Context:
This problem tests the use of algebraic identities with surds (square roots) and symmetric expressions in two variables x and y. Rather than solving directly for x and y, we use identities such as (x + y)^2 and (x - y)^2 to derive expressions for xy and x^2 + y^2. This is a powerful method in competitive exams, where direct solving may be lengthy or unnecessary, but cleverly using identities leads to a quick and elegant solution.

Given Data / Assumptions:
- x + y = sqrt(3).
- x - y = sqrt(2).
- We need to compute 8 * x * y * (x^2 + y^2).
- Square roots are exact; we should not approximate them with decimals.

Concept / Approach:
The approach uses two main identities:
1) (x + y)^2 = x^2 + 2xy + y^2.
2) (x - y)^2 = x^2 - 2xy + y^2.
By adding and subtracting these equations, we can find xy and x^2 + y^2 in terms of the given square roots. Once we know xy and x^2 + y^2, we substitute them into the expression 8xy(x^2 + y^2) and simplify to a rational number.

Step-by-Step Solution:
Step 1: Compute (x + y)^2 using the given value: (x + y)^2 = (sqrt(3))^2 = 3.Step 2: Write (x + y)^2 in identity form: x^2 + 2xy + y^2 = 3.Step 3: Compute (x - y)^2 using the given value: (x - y)^2 = (sqrt(2))^2 = 2.Step 4: Write (x - y)^2 in identity form: x^2 - 2xy + y^2 = 2.Step 5: Subtract the second equation from the first: (x^2 + 2xy + y^2) - (x^2 - 2xy + y^2) = 3 - 2.Step 6: The left side becomes 4xy, and the right side is 1, so 4xy = 1, which gives xy = 1 / 4.Step 7: Now add the two equations: (x^2 + 2xy + y^2) + (x^2 - 2xy + y^2) = 3 + 2.Step 8: The left side becomes 2x^2 + 2y^2 = 5, so x^2 + y^2 = 5 / 2.Step 9: Substitute into the target expression: 8xy(x^2 + y^2) = 8 * (1 / 4) * (5 / 2).Step 10: Simplify: 8 * (1 / 4) = 2, and 2 * (5 / 2) = 5.Step 11: Therefore, 8xy(x^2 + y^2) = 5.

Verification / Alternative check:
We can also attempt to find x and y explicitly. From x + y = sqrt(3) and x - y = sqrt(2), add the equations to get 2x = sqrt(3) + sqrt(2), so x = (sqrt(3) + sqrt(2)) / 2. Subtract to get 2y = sqrt(3) - sqrt(2), so y = (sqrt(3) - sqrt(2)) / 2. Substituting these into xy and x^2 + y^2 gives the same results derived earlier, confirming the value 5 for the given expression.

Why Other Options Are Wrong:
- 6 and 4: These are simple integers and might result from miscomputing xy or x^2 + y^2, or from using incorrect coefficients in the identities.
- sqrt(6) and sqrt(5): These suggest that the student left surds in the final answer instead of simplifying fully, or did not use symmetric identities properly. The final result here is a rational integer, not a surd.

Common Pitfalls:
A common error is to attempt to find x and y numerically right away, which can be time consuming and prone to mistakes. Another pitfall is mixing up the identities for (x + y)^2 and (x - y)^2 or forgetting that subtracting them gives 4xy. Some students incorrectly treat sqrt(3) and sqrt(2) as approximate decimals and introduce rounding errors, which is unnecessary because the final answer is a clean integer. Using exact algebraic manipulation is both faster and safer in exam conditions.

Final Answer:
The value of the expression 8 * x * y * (x^2 + y^2) is 5.