Difficulty: Medium
Correct Answer: 280
Explanation:
Introduction / Context:
This question tests using algebraic identities to evaluate expressions involving a and b when their sum and product are given. Instead of solving for a and b individually (which is possible but unnecessary), you can compute a^3 + b^3 directly using the identity:
a^3 + b^3 = (a + b)^3 − 3ab(a + b).
This identity is especially useful in aptitude problems because it transforms a higher-power expression into something that depends only on (a + b) and ab, both of which are already provided. The main challenge is applying the identity correctly and handling arithmetic accurately when expanding (a + b)^3 and multiplying 3ab(a + b).
Given Data / Assumptions:
Concept / Approach:
Compute (a + b)^3 using the given sum. Then compute 3ab(a + b) using the given product and sum. Subtract the second from the first to obtain a^3 + b^3. This avoids quadratic solving and gives the value in a few steps.
Step-by-Step Solution:
1) Use the identity:
a^3 + b^3 = (a + b)^3 − 3ab(a + b)
2) Compute (a + b)^3:
(a + b)^3 = 10^3 = 1000
3) Compute 3ab(a + b):
3ab(a + b) = 3 * 24 * 10 = 720
4) Subtract:
a^3 + b^3 = 1000 − 720 = 280
Verification / Alternative check:
Optional check by solving: a and b would be roots of t^2 − 10t + 24 = 0. This factors as (t − 6)(t − 4) = 0, so a and b are 6 and 4 in some order. Then a^3 + b^3 = 6^3 + 4^3 = 216 + 64 = 280, matching the identity-based result exactly. This confirms the computation is correct.
Why Other Options Are Wrong:
• 152 and 140: usually come from subtracting 3ab only (72) instead of 3ab(a + b) (720).
• 72: equals 3ab, not a^3 + b^3.
• 328: indicates an arithmetic error in 10^3 or in 3*24*10.
Common Pitfalls:
• Using a^3 + b^3 = (a + b)^3 − 3ab (missing the factor (a + b)).
• Computing 10^3 incorrectly or mis-multiplying 3*24*10.
• Trying to find a and b first and making factoring mistakes, even though it is not needed.
Final Answer:
280
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