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

Verified step by step guidance

1. Recognize that finding the complex cube roots of \(-1\) involves solving the equation \(z^3 = -1\).
2. Express \(-1\) in polar form as \(1(\cos(\pi) + i\sin(\pi))\).
3. Use De Moivre's Theorem, which states that if \(z = r(\cos(\theta) + i\sin(\theta))\), then the \(n\)-th roots are given by \(z_k = r^{1/n}(\cos((\theta + 2k\pi)/n) + i\sin((\theta + 2k\pi)/n))\) for \(k = 0, 1, \ldots, n-1\).
4. Calculate the cube roots by setting \(n = 3\), \(r = 1\), and \(\theta = \pi\), and find the three roots: \(z_0, z_1, z_2\).
5. Convert each root from polar form back to rectangular form using \(z_k = \cos(\theta_k) + i\sin(\theta_k)\) for each \(k\).

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 and b are real numbers, and i is the imaginary unit defined as the square root of -1. Understanding complex numbers is essential for solving problems involving roots of negative numbers, as they allow for the extension of the number system beyond real numbers.

Roots of Complex Numbers

Finding the roots of complex numbers involves determining the values that, when raised to a certain power, yield the original complex number. For cube roots, we seek three distinct values that satisfy the equation z^3 = -1. This process often utilizes polar coordinates and De Moivre's Theorem to simplify calculations.

Rectangular Form

Rectangular form refers to expressing complex numbers in the standard a + bi format, where a is the real part and b is the imaginary part. When finding roots of complex numbers, it is important to convert any results from polar or exponential forms back into rectangular form for clarity and ease of interpretation, especially in applications involving real-world problems.