If z is a non-real root of z^7+1=0, then z^86+z^175+z^289 is equal to (a) 0 (b) -1 (c) 3 (d) 1

Name: Problem 53, Chapter 1: IIT JEE Super Course in Mathematics ALGEBRA II
Uploaded: 2021-06-05T09:40:23-08:00
Duration: 3 min 29 s
Channel: Anas Venkitta
Description: If z is a non-real root of z^7+1=0, then z^86+z^175+z^289 is equal to (a) 0 (b) -1 (c) 3 (d) 1

step 1 In the given question we are told that if z is a root of a non -real root, that is a complex roo

If z is a non-real root of z^7+1=0, then z^86+z^175+z^289 is...

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Key Concepts

Roots of Unity

Roots of unity are complex numbers that satisfy the equation z^n = 1 for some positive integer n. They are evenly spaced on the unit circle in the complex plane and possess the property that they repeat cyclically when raised to higher powers. This concept is fundamental in complex number theory, Fourier analysis, and various areas of algebra.

Cyclic Properties of Exponential Functions

The cyclic nature of the exponential function in the context of complex numbers, particularly when dealing with roots of unity, means that powers of these numbers repeat in a fixed cycle. This property allows for simplification of expressions involving high exponents by reducing them modulo the order of the root of unity.

Modular Exponentiation in Complex Numbers

Modular exponentiation in the context of complex numbers involves reducing large exponents by the period of the cyclic group formed by the roots of unity. By calculating exponents modulo the order of the root (for instance, mod n when dealing with n-th roots of unity), complex expressions can be greatly simplified, which is especially useful in solving polynomial equations with complex roots.

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