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Let α and β be the roots of the quadratic equation x² - x + 1 = 0. What is the value of α²⁰⁰⁹ + β²⁰⁰⁹?

1. −1
2. 1
3. 0
4. −2

Correct answer: 1

Solution

The roots α and β of the equation x² - x + 1 = 0 can be expressed in terms of complex numbers, specifically as α = e^(iπ/3) and β = e^(-iπ/3). Using De Moivre's theorem, we find that αⁿ + βⁿ can be computed using the periodicity of complex exponentials, leading to α²⁰⁰⁹ + β²⁰⁰⁹ = 1.

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