In Exercises 32–35, find all the complex roots. Write roots in rectangular form. The complex fifth roots of −1 − i

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part and 'b' is the imaginary part. In this context, -1 - i is a complex number where -1 is the real part and -1 is the coefficient of the imaginary unit 'i'. Understanding how to manipulate and represent complex numbers is essential for finding their roots.

Polar Form of Complex Numbers

The polar form of a complex number expresses it in terms of its magnitude (r) and angle (θ) with respect to the positive real axis, represented as r(cos θ + i sin θ) or re^(iθ). This form is particularly useful for finding roots, as it simplifies the process of applying De Moivre's Theorem, which states that the nth roots of a complex number can be found by dividing the angle by n and taking the nth root of the magnitude.

De Moivre's Theorem

De Moivre's Theorem provides a method for raising complex numbers in polar form to a power or extracting roots. It states that for a complex number in polar form r(cos θ + i sin θ), the nth roots can be calculated as r^(1/n)(cos(θ/n + 2kπ/n) + i sin(θ/n + 2kπ/n)), where k is an integer from 0 to n-1. This theorem is crucial for determining the complex fifth roots of -1 - i.