Problem 62 - A Graphical Approach to College Algebra

Question

Find all \(n\) complex solutions of each equation of the form \(x^{n}=k\) $$x^{4}=81$$

Step-by-Step Solution

Answer

The solutions are: 3, 3i, -3, -3i.

Step 1: Express in Polar Form

First, recognize that 81 can be expressed in polar form. Since 81 is a positive real number, we can express it as 81 with an argument of 0: \[ 81 = 81 \text{cis}(0) \] where \(\text{cis}(\theta) = \cos(\theta) + i\sin(\theta)\).

Step 2: Apply De Moivre's Theorem

According to De Moivre's Theorem, the complex solutions of the equation \(x^4 = 81\) are given by: \[ x = 81^{1/4} \text{cis}\left(\frac{0 + 2k\pi}{4}\right) \] where \(k = 0, 1, 2, 3\), representing the four roots.

Step 3: Calculate the Magnitude

Calculate the magnitude of the solutions. The fourth root of the magnitude 81 is: \[ 81^{1/4} = 3 \] So the magnitude of the complex number is 3.

Step 4: Determine the Argument for Each Root

Calculate the argument of each root using the formula for \(k\):- For \(k=0\), the argument is \(\frac{2 \pi k}{4} = 0\).- For \(k=1\), the argument is \(\frac{2 \pi}{4} = \frac{\pi}{2}\).- For \(k=2\), the argument is \(\frac{4 \pi}{4} = \pi\).- For \(k=3\), the argument is \(\frac{6 \pi}{4} = \frac{3\pi}{2}\).

Step 5: Write the Solutions

Combine the magnitude and arguments to express the solutions:- For \(k=0\): \[ x_0 = 3 \text{cis}(0) = 3 \]- For \(k=1\): \[ x_1 = 3 \text{cis}\left(\frac{\pi}{2}\right) = 3i \]- For \(k=2\): \[ x_2 = 3 \text{cis}(\pi) = -3 \]- For \(k=3\): \[ x_3 = 3 \text{cis}\left(\frac{3\pi}{2}\right) = -3i \]

Step 6: Conclude the Solution Set

The four complex solutions of the equation \(x^4 = 81\) are: - \(3\), - \(3i\), - \(-3\), - \(-3i\).

Key Concepts

De Moivre's Theorem

De Moivre's Theorem is a powerful tool in complex number theory. It allows us to find powers and roots of complex numbers when they are expressed in polar form. This theorem states that for any complex number, expressed as \(r \text{cis}(\theta)\), its nth power is given by: \[ (r \text{cis}(\theta))^n = r^n \text{cis}(n\theta) \] where \(r\) is the magnitude and \(\theta\) is the argument (angle). This is particularly useful not just for multiplying but also for finding roots of equations involving complex numbers. By breaking the complex number down into its magnitude and angle, we can easily manipulate them using simple multiplication and addition.

Polar Form

The polar form of a complex number is an alternative way to express a complex number using its magnitude and angle. Rather than using the standard \(a + bi\) form, where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit, polar form uses \(r \text{cis}(\theta)\). In this form, \(r\) is the magnitude or modulus of the complex number and \(\theta\) is the argument or angle with the positive real axis.

• Magnitude (\(r\)) can be found using \(r = \sqrt{a^2 + b^2}\).
• Argument (\(\theta\)) is the angle from the positive real axis, found using \(\theta = \tan^{-1}(\frac{b}{a})\).

Polar form is especially useful in multiplication and division of complex numbers, simplifying the process to simple arithmetic rather than algebra.

Complex Solutions

Complex solutions arise when solving equations with real or complex coefficients where solutions are not solely real numbers. In many cases, complex solutions will appear in conjugate pairs, but they can also be represented in polar form using De Moivre's Theorem. The example equation \(x^4 = 81\) demonstrates this concept well. We find the fourth roots of 81 by expressing the number in polar form and utilizing the power of angles to derive multiple solutions. By recognizing that complex solutions can exist in multiple dimensions of the complex plane, we adopt a fuller understanding of the intricate patterns that complex numbers form.

Roots of Unity

Roots of unity are a fascinating concept linked closely to complex numbers and De Moivre's Theorem. They are solutions to the equation \(x^n = 1\). In essence, these are complex numbers that, when raised to a positive integer \(n\), result in 1. These roots are evenly spaced on the unit circle in the complex plane. For the equation \(x^4 = 81\), the roots of unity are crucial in understanding why there are four distinct solutions. Using De Moivre's, the general formula for an nth root of unity is: \[ x = \text{cis}(\frac{2\pi k}{n}), \quad k = 0, 1, 2, \ldots, n-1 \] where each distinct angle represents a unique solution. This reveals the cyclic nature of these roots as they exhibit symmetries in the complex plane, beautifully illustrating a fundamental aspect of complex analysis.