If is a non-real complex cube root of unity
MATHEMATICS · Multi-part · CBSE · Mathematics · Complex Numbers and Quadratic Equations · Complex Numbers and Quadratic Equations
Problem
If ω is a non-real complex cube root of unity, find the values of the following.
(i) \omega^{1999}
(ii) \omega^{-998}
(iii) \left( \frac{-1 + i\sqrt{3}}{2} \right)^{3n+2}, n \in \mathbb{N} \text{ and } i = \sqrt{-1}
(iv) (1 + \omega)(1 + \omega^2)(1 + \omega^4)(1 + \omega^8) \dots \text{ upto } 2n \text{ factors}
(v) \left( \frac{\alpha + \beta\omega + \gamma\omega^2 + \delta\omega^3}{\beta + \alpha\omega^2 + \gamma\omega + \delta\omega^3} \right), \text{where } \alpha, \beta, \gamma, \delta \in \mathbb{R}
(vi) 1 \cdot (2 - \omega)(2 - \omega^2) + 2 \cdot (3 - \omega)(3 - \omega^2) + 3 \cdot (4 - \omega)(4 - \omega^2) + \dots + (n-1) \cdot (n - \omega)(n - \omega^2)
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