If αand βare the complex cube roots of unity, then α^7+β^7= (a) (1)/(αβ) (b) (-1)/(αβ) (c) αβ(d)

step 1 So in the given question we are told that alpha and beta are the complex cube roots of 1. Comple

If αand βare the complex cube roots of unity, then α^7+β^7=...

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Key Concepts

Complex Roots of Unity

Complex roots of unity are the solutions to the equation x^n = 1 in the complex number plane. Specifically, for any positive integer n, these roots lie on the unit circle and are separated by equal angles. They exhibit deep symmetry, which is fundamental in various areas of mathematics including algebra and number theory.

Modular Exponentiation with Roots of Unity

The exponents of roots of unity can be reduced modulo n because the roots exhibit a cyclic group structure under multiplication. This means that when raising a complex root of unity to a high power, one can simplify the exponent by taking it modulo n, where n is the order of the root. This process is key in simplifying expressions and evaluating sums involving powers of these roots.

Symmetric Sum Properties

The roots of unity have symmetric properties that result from their arrangement on the unit circle. For instance, the sum of all nth roots of unity is zero, and sums involving specific subsets of these roots (like the non-real ones) can often be expressed in elegant and simple forms. These symmetric properties are utilized to simplify complex expressions and solve equations efficiently.

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