Let's begin by solving **\( \sqrt[5]{1} \)**, which represents the fifth roots of unity. These roots lie on the unit circle in the complex plane.

### Step-by-Step Explanation:
The equation \( z^5 = 1 \) has five solutions, known as the fifth roots of unity. These solutions are spaced equally around the unit circle, with each root corresponding to a complex number of modulus 1 and different arguments.

#### Formula for Roots of Unity:
To find these roots, we use the general formula for the \( n \)-th roots of unity:
\[
z_k = e^{2\pi i k / n} \quad \text{for} \quad k = 0, 1, 2, \ldots, n-1
\]
where \( n = 5 \) (fifth root), and \( z_k \) are the different roots.

For \( \sqrt[5]{1} \), the five roots are:

\[
z_k = e^{2\pi i k / 5} \quad \text{for} \quad

Solve the fifth root of unity, \( \sqrt[5]{1} \).

Explore the step-by-step solution of the fifth roots of unity, \( \sqrt[5]{1} \), using complex numbers and the roots of unity theorem. Learn how to find the five solutions on the unit circle. Suitable for college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation