Two numbers a and b satisfy a - b = 2 and ab = 24. Without finding a and b individually, use algebraic identities to compute the exact value of a^3 - b^3 and select the correct option.

Introduction / Context:
This simplification question checks your ability to use algebraic identities cleverly so that you avoid explicitly solving a quadratic equation for a and b. Instead of finding the two numbers directly, you use the given information about their difference and product to compute a^3 - b^3 efficiently, which is a valuable skill in competitive exams where time is limited.

Given Data / Assumptions:

• a - b = 2.
• ab = 24.
• a and b are real numbers.
• We need to find a^3 - b^3 exactly.
• We are encouraged to use standard algebraic identities rather than solving for a and b explicitly.

Concept / Approach:
The identity for the difference of cubes is a^3 - b^3 = (a - b)(a^2 + ab + b^2). However, we do not know a^2 + ab + b^2 directly. Instead of solving for a and b, we can express a^3 - b^3 in terms of a - b and ab using a rearranged identity: a^3 - b^3 = (a - b)^3 + 3ab(a - b). This version uses only a - b and ab, both of which are given. Once we plug in those values and compute carefully, we reach the final answer in a few steps.

Step-by-Step Solution:
Recall the identity: a^3 - b^3 = (a - b)^3 + 3ab(a - b). Given a - b = 2 and ab = 24, substitute into the formula. Compute (a - b)^3 = 2^3 = 8. Compute 3ab(a - b) = 3 * 24 * 2 = 144. Add the two parts: a^3 - b^3 = 8 + 144 = 152.
Verification / Alternative check:
As a verification, you may solve for a and b explicitly. From a - b = 2, we get a = b + 2. Substitute this into ab = 24 to obtain b(b + 2) = 24, which simplifies to b^2 + 2b - 24 = 0. Solving gives b = 4 or b = -6. Correspondingly, a = 6 or a = -4. For the pair (a, b) = (6, 4), a^3 - b^3 = 216 - 64 = 152. For the pair (-4, -6), we get (-4)^3 - (-6)^3 = -64 - (-216) = 152 again, confirming the identity based result.

Why Other Options Are Wrong:
Options b (140), c (124), and d (280) may arise from mistakes such as forgetting the 3 factor in 3ab(a - b), using ab(a - b) instead, or squaring instead of cubing. Option e (168) could come from miscomputing 3 * 24 * 2 as 160 or 168 through arithmetic errors. Only 152 matches correct use of the identity and accurate multiplication.

Common Pitfalls:
A frequent error is to start by computing a^2 + ab + b^2 separately, which can be more time consuming if done via separate identities. Another pitfall is misremembering the cube expansion or mixing up a^3 - b^3 with a^3 + b^3. Carefully recalling and applying the exact formula a^3 - b^3 = (a - b)^3 + 3ab(a - b) avoids these mistakes and leads directly to the solution.

Final Answer:
The exact value of a^3 - b^3, given a - b = 2 and ab = 24, is 152.