In algebraic identities, you are given that a^3 + b^3 = 28 and a + b = 4 for real numbers a and b. Using a standard identity for a^3 + b^3, find the exact value of ab.

Introduction / Context:This question checks whether you can connect a sum of cubes with the sum of numbers using the identity for a^3 + b^3. It is a common simplification and identity-based technique in aptitude algebra.

Given Data / Assumptions:

• a^3 + b^3 = 28
• a + b = 4
• a and b are real numbers

Concept / Approach:Use the identity: a^3 + b^3 = (a + b)^3 - 3ab(a + b). Since a^3 + b^3 and (a + b) are given, this identity becomes a direct equation in ab.

Step-by-Step Solution:

Verification / Alternative check:If ab = 3 and a + b = 4, then a and b are roots of t^2 - 4t + 3 = 0, giving t = 1 and 3. Then a^3 + b^3 = 1^3 + 3^3 = 28, which matches perfectly.

Why Other Options Are Wrong:

Common Pitfalls:Forgetting that the identity is (a + b)^3 - 3ab(a + b), not (a + b)^3 - 3ab. Also, many learners forget to multiply by (a + b) in the last term.

Final Answer:ab = 3