In Exercises 81–86, solve each equation in the complex number system. Express solutions in polar and rectangular form.x⁶ − 1 = 0

Verified step by step guidance

1. Recognize that the equation \(x^6 - 1 = 0\) can be rewritten as \(x^6 = 1\).
2. Identify that \(x^6 = 1\) represents the sixth roots of unity, which are complex numbers evenly spaced on the unit circle in the complex plane.
3. Express the number 1 in polar form as \(1 = e^{i(2k\pi)}\) where \(k\) is an integer.
4. Find the sixth roots by solving \(x = e^{i(\frac{2k\pi}{6})}\) for \(k = 0, 1, 2, 3, 4, 5\).
5. Convert each solution from polar form \(e^{i\theta}\) to rectangular form using \(x = \cos(\theta) + i\sin(\theta)\).

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. They can be represented on the complex plane, with the x-axis as the real axis and the y-axis as the imaginary axis. Understanding complex numbers is essential for solving equations that do not have real solutions.

Polar Form of Complex Numbers

The polar form of a complex number expresses it in terms of its magnitude (r) and angle (θ), represented as r(cos θ + i sin θ) or re^(iθ). This form is particularly useful for multiplication and division of complex numbers and for solving equations involving roots, as it simplifies the process of finding solutions in the complex plane.

Roots of Unity

Roots of unity are complex numbers that satisfy the equation z^n = 1, where 'n' is a positive integer. These roots are evenly spaced around the unit circle in the complex plane and can be expressed in polar form as e^(2πik/n) for k = 0, 1, ..., n-1. In the context of the given equation x⁶ - 1 = 0, the sixth roots of unity will provide the solutions in both polar and rectangular forms.