Difficulty: Medium
Correct Answer: 19
Explanation:
Introduction / Context:
This algebra question asks you to compute a^3 − b^3 given the values of a − b and ab. Instead of trying to solve explicitly for a and b, you can use an identity that links a^3 − b^3 directly with a − b and ab. Such identities are especially useful in exams because they avoid unnecessary quadratic solving and keep calculations manageable.
Given Data / Assumptions:
Concept / Approach:
The key identity is a^3 − b^3 = (a − b)(a^2 + ab + b^2). We already know a − b, so we need to find a^2 + ab + b^2. Another useful identity is (a − b)^2 = a^2 + b^2 − 2ab, which allows us to express a^2 + b^2 in terms of a − b and ab. With a^2 + b^2 known, we can add ab to get a^2 + ab + b^2, and finally multiply by a − b.
Step-by-Step Solution:
Verification / Alternative check:
As an alternative, you could solve for a and b explicitly using the system a − b = 1 and ab = 6. This leads to a quadratic equation in a or b whose roots will satisfy these conditions. Computing a^3 − b^3 using the actual numerical roots would confirm that the result is indeed 19, although it takes more time than using the identity directly.
Why Other Options Are Wrong:
The values 21, 23, 25, and 17 would arise from miscalculating a^2 + b^2 or ab, or from misapplying the cube identity. For example, forgetting that (a − b)^2 includes −2ab or using a^3 + b^3 instead of a^3 − b^3 could easily lead to these incorrect numbers. Only 19 is consistent with the correct identity based derivation.
Common Pitfalls:
Common mistakes include confusing the formula for (a + b)^2 with that for (a − b)^2, or forgetting to add ab when forming a^2 + ab + b^2. Some learners also attempt to expand a^3 − b^3 directly, which is more cumbersome than using the identity. Keeping these standard identities well memorised and understood makes such problems much easier.
Final Answer:
The value of a^3 − b^3, given a − b = 1 and ab = 6, is 19.
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