Problem 61 - A Graphical Approach to Precalculus with Limits

Question

Find all \(n\) complex solutions of each equation of the form \(x^{n}=k\) $$x^{4}=16$$

Step-by-Step Solution

Answer

The complex solutions are \(\sqrt{2}, i\sqrt{2}, -\sqrt{2}, \text{ and } -i\sqrt{2}\).

Step 1: Identify powers and constant

Given the equation is \(x^4 = 16\). Here, we need to find the fourth roots of 16. Recognize that 16 is equivalent to \(16 = 2^4\).

Step 2: Express in exponential form

Convert the equation into exponential form: \(x^4 = (2e^{0i})^4\). Notice that 16 has a modulus of 2 and an argument (angle) of 0 radians in the complex plane.

Step 3: Use De Moivre's Theorem

According to De Moivre's Theorem, the roots are given by \(x_k = \, r^{1/4}\, \textcolor{white}{,} e^{i(\theta + 2k\pi)/n}\) where \(r = 2\), \(\theta = 0\), \(n = 4\), and \(k = 0, 1, 2, 3\).

Step 4: Calculate each root

Calculate the four distinct roots using the formula: - For \(k = 0\): \(x_0 = 2^{1/4} e^{i(0 + 0\cdot\frac{2\pi}{4})} = \sqrt{2}\)- For \(k = 1\): \(x_1 = 2^{1/4} e^{i(0 + 1\cdot\frac{2\pi}{4})} = \sqrt{2} e^{\frac{i\pi}{2}} = i\sqrt{2}\)- For \(k = 2\): \(x_2 = 2^{1/4} e^{i(0 + 2\cdot\frac{2\pi}{4})} = \sqrt{2} e^{i\pi} = -\sqrt{2}\)- For \(k = 3\): \(x_3 = 2^{1/4} e^{i(0 + 3\cdot\frac{2\pi}{4})} = \sqrt{2} e^{\frac{3i\pi}{2}} = -i\sqrt{2}\)

Step 5: Summarize the roots

The solutions to the equation \(x^4 = 16\) are: \(\sqrt{2}, i\sqrt{2}, -\sqrt{2}, \text{ and } -i\sqrt{2}\). These represent the fourth roots of 16 in the complex plane.

Key Concepts

De Moivre's Theorem

De Moivre's Theorem is a powerful tool in complex analysis that makes finding powers and roots of complex numbers much easier. Simply put, it states that if we have a complex number \(z = r(cos(\theta) + i \sin(\theta))\), raising \(z\) to the \(n\)-th power, or taking the \(n\)-th root, can be done using the expression:

• Power: \(z^n = r^n \cdot ( \cos(n\theta) + i \sin(n\theta) )\)
• Root: \(z^{1/n} = r^{1/n} \cdot ( \cos(\frac{\theta + 2\pi k}{n}) + i \sin(\frac{\theta + 2\pi k}{n}) )\), where \(k = 0, 1, 2, ..., n-1\) are the possible roots

This theorem essentially breaks down complex numbers into their polar form, making calculations more straightforward. For example, in the exercise above, we expressed \(16 = 2^4\) in polar form and applied De Moivre's Theorem to compute the fourth roots. Understanding and applying this theorem helps simplify finding solutions to equations like \(x^4 = 16\) in the complex plane.

Fourth Roots

The concept of fourth roots is pertinent when finding solutions to equations like \(x^4 = 16\). A fourth root of a number \(k\) means any number \(x\) that satisfies \(x^4 = k\). For real numbers, the fourth root is related to the square root, but for complex numbers, it involves a broader perspective in the complex plane.

To find these roots, we essentially solve an equation of the form \(x^n = k\) where \(n\) is 4, using De Moivre's Theorem. We convert the problem into one involving angles and the unit circle:

• Set the modulus and angle of \(k\), for example, converting \(16\) to \(2e^{0i}\)
• Calculate \(x_k = r^{1/4} e^{i(\frac{\theta + 2\pi k}{n})}\), with \(k\) ranging from 0 to 3
• Obtain results like \((\sqrt{2}, i\sqrt{2}, -\sqrt{2}, -i\sqrt{2})\), showcasing roots' symmetry around the complex plane

Fourth roots often reveal the intriguing symmetry and beauty inherent in complex numbers. They show how solutions revolve around the origin in a circle, evenly spaced, an important concept when dealing with complex solutions.

Complex Plane

Understanding the complex plane is crucial when dealing with complex solutions. Often described as the Argand plane, it visually represents complex numbers, where the x-axis is the real part, and the y-axis is the imaginary part.

Imagine plotting a point \(a + bi\) on this plane. Here, \(a\) is the real axis, and \(b\) is the imaginary axis. This representation helps you grasp the behavior of complex functions and operations on them, such as addition, multiplication, or finding roots:

• Complex numbers can be visualized as vectors emanating from the origin
• De Moivre's Theorem operates using the angle (or argument) each vector makes with the real axis
• The operation of taking roots means finding vectors that are evenly spaced around the unit circle at the given radius \(r^{1/n}\)

In the example provided, finding fourth roots of 16 meant picking points that form a square around the origin because each root represents a 90-degree rotation from the previous, an integral concept in understanding the layout of solutions.