In Exercises 69–76, find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest tenth. The complex fifth roots of 32 (cos 5π/3 + i sin 5π/3)

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 coefficient of the imaginary unit 'i'. Understanding complex numbers is essential for solving problems involving roots, especially in trigonometric contexts where angles and magnitudes are involved.

De Moivre's Theorem

De Moivre's Theorem states that for any complex number in polar form r(cos θ + i sin θ), the nth roots can be found using the formula 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 finding complex roots, as it simplifies the process of extracting roots from complex numbers.

Rectangular Form

Rectangular form refers to the standard way of expressing complex numbers as a + bi. When finding complex roots, it is often necessary to convert from polar form (which uses magnitude and angle) to rectangular form for clarity and ease of interpretation. This conversion involves calculating the real and imaginary components based on the cosine and sine of the angle.