Difficulty: Medium
Correct Answer: – 1
Explanation:
Introduction / Context:The relation a + 1/a = 1 leads to a quadratic equation in a. From that, powers of a can be reduced systematically. Here we are asked for a^3, which can be found without computing any decimals by using the quadratic to eliminate higher powers.
Given Data / Assumptions:
Concept / Approach:Multiply the given equation by a to form a quadratic: a^2 − a + 1 = 0. Then express a^2 in terms of a and substitute into a^3 = a*a^2 to reduce a^3 to a linear expression in a, which then collapses to a constant using the same quadratic again.
Step-by-Step Solution:
From a + 1/a = 1 ⇒ multiply by a: a^2 + 1 = a ⇒ a^2 − a + 1 = 0.Hence a^2 = a − 1.Compute a^3 = a * a^2 = a(a − 1) = a^2 − a.Replace a^2 using a^2 − a + 1 = 0 ⇒ a^2 = a − 1.Thus a^3 = (a − 1) − a = −1.Verification / Alternative check:
The roots of a^2 − a + 1 = 0 are complex conjugates with magnitude 1; their cube equals −1, consistent with the algebraic reduction.Why Other Options Are Wrong:
Common Pitfalls:
Final Answer:
– 1
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