Introduction / Context:
This question again focuses on the algebraic identity for the difference of cubes. Instead of finding the actual values of a and b from the given conditions, you are expected to use identities that express a^3 - b^3 in terms of a - b and ab. This is a powerful technique because it saves time and reduces the chance of algebraic errors that can occur when solving quadratics under exam pressure.
Given Data / Assumptions:
- a - b = 1.
- ab = 6.
- a and b are real numbers.
- The task is to find a^3 - b^3.
- You should use identities rather than solving explicitly for a and b.
Concept / Approach:
The relevant identity is a^3 - b^3 = (a - b)^3 + 3ab(a - b). This identity is derived by expanding (a - b)^3 and then rearranging terms. Because the given information includes a - b and ab, this identity allows us to substitute directly and compute a^3 - b^3 in just a few steps. This approach is especially suited to quantitative aptitude questions where speed and accuracy are critical.
Step-by-Step Solution:
Use the identity: a^3 - b^3 = (a - b)^3 + 3ab(a - b).
Substitute a - b = 1 and ab = 6.
Compute (a - b)^3 = 1^3 = 1.
Compute 3ab(a - b) = 3 * 6 * 1 = 18.
Add them: a^3 - b^3 = 1 + 18 = 19.
Verification / Alternative check:
To verify, you can solve for a and b explicitly, although this is not necessary in an exam. From a - b = 1, we have a = b + 1. Substitute into ab = 6 to get b(b + 1) = 6, which simplifies to b^2 + b - 6 = 0. Solving gives b = 2 or b = -3. If b = 2, then a = 3, and a^3 - b^3 = 27 - 8 = 19. If b = -3, then a = -2, and a^3 - b^3 = (-8) - (-27) = 19. Both pairs confirm the identity based result.
Why Other Options Are Wrong:
Options b (21), c (23), d (25), and e (27) can arise from partial application of the identity or incorrect arithmetic. For example, using 3ab instead of 3ab(a - b), or miscomputing 3*6 as 16, will lead to such incorrect results. None of these values match both the identity calculation and the explicit check using actual values of a and b.
Common Pitfalls:
A common mistake is to use the sum of cubes identity a^3 + b^3 by accident, or to misremember the formula for the difference of cubes. Another error is to attempt to compute a^2 + ab + b^2 separately using additional identities, which increases the risk of mistakes. Staying with the compact identity a^3 - b^3 = (a - b)^3 + 3ab(a - b) and substituting carefully is the most reliable and efficient method.
Final Answer:
The exact value of a^3 - b^3, given a - b = 1 and ab = 6, is
19.
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