Problem 84 - Precalculus with Limits

Question

Finding the \(n\) th Roots of a Complex Number \(\operatorname{In}\) Exercises \(81-96,(\) a) use the formula on page 446 to find the indicated roots of the complex number, (b) represent each of the roots graphically, and (c) write each of the roots in standard form. Fifth roots of $$32\left(\cos \frac{5 \pi}{6}+i \sin \frac{5 \pi}{6}\right)$$

Step-by-Step Solution

Answer

The fifth roots of the given complex number are two complex numbers in polar form and their corresponding standard forms. These roots will form a pentagon when plotted on the complex plane due to their equal separation of 72 degrees.

Step 1: Calculate The Magnitude of the Complex Number

The magnitude of the complex number is calculated by raising the given number (32) to the power of 1/n where n is 5 (since we're finding the fifth roots). This gives: \(32^{\frac{1}{5}} = 2\)

Step 2: Calculate The Arguments of the Roots

The arguments of the roots can be found using the formula: \(\theta + \frac{2k\pi}{5}\) where \(\theta = \frac{5\pi}{6}\) and \(k = 0, 1, 2, 3, 4\). Calculate the arguments with these values of \(k\), giving 5 results.

Step 3: Convert The Roots To Polar Form

Each root will be in the form of: \(2(\cos(\theta) + i \sin(\theta))\), where \(\theta\) are the arguments calculated in the previous step. Calculate to get the 5 roots.

Step 4: Represent Roots Graphically

Plotting the roots will show them as points on the complex plane at angles of 72 degrees apart and a radius of 2 (the magnitude). Use the polar coordinates of the roots to plot these points.

Step 5: Convert to Standard Form

This step involves converting the roots from polar form to standard form (\(a+bi\)). Use the identities \(\cos(\theta) = \frac{e^{i\theta} + e^{-i\theta}}{2}\) and \(\sin(\theta) = \frac{e^{i\theta} - e^{-i\theta}}{2i}\) to convert the roots to standard form.

Key Concepts

Nth Roots of Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. When we talk about the "nth roots" of a complex number, we are referring to finding all of the complex numbers that, when raised to the power of n, will equal our original complex number. This exercise deals with finding fifth roots, meaning we want to find all the complex numbers that give us the original number when raised to the fifth power.

To find the nth roots, you'll use a formula that takes into account both the magnitude and the angle of the original complex number. Here's how it generally works:

• Calculate the magnitude or length of the number in polar form. For example, when n is 5, you'd find the fifth root of the given magnitude.
• Determine the angles of each root. Use the theory of rotational symmetry; each successive root is rotated by the same angle in the complex plane.

Understanding nth roots helps visualize complex numbers and appreciate the geometry underlying complex number calculations.

Polar Form of Complex Numbers

The polar form of a complex number expresses the number using its magnitude and angle relative to the real axis. This form is written as: \[ r(\cos(\theta) + i\sin(\theta)) \] where \( r \) represents the magnitude (usually a positive real number), and \( \theta \) is the angle or argument.

To convert a complex number from its standard form \( a + bi \) to polar form, follow these steps:

• Calculate the magnitude, \( r = \sqrt{a^2 + b^2} \).
• Find the argument, \( \theta = \tan^{-1}(\frac{b}{a}) \).

Polar form is particularly useful for multiplication and finding powers and roots, as multiplying complex numbers involves simply adding their angles while keeping the magnitudes in check.

Standard Form of Complex Numbers

Standard form is a simple way to express complex numbers. It's the form \( a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part. Each complex number can be uniquely represented in this way, making calculations straightforward.

To convert from polar to standard form, use the identities for cosine and sine:

• The real part \( a \) is \( r\cos(\theta) \).
• The imaginary part \( b \) is \( r\sin(\theta) \).

Hence, converting gives \( a + bi = r\cos(\theta) + r\sin(\theta)i \). Understanding standard form is essential for basic arithmetic operations on complex numbers and provides a concrete way of expressing them for further computation.

Graphically Representing Complex Numbers

Graphically, complex numbers are represented on the complex plane, where the horizontal axis (x-axis) represents the real part, and the vertical axis (y-axis) represents the imaginary part.

For example, the complex number \( a + bi \) is plotted as the point \((a, b)\) in this plane. This visual representation helps in understanding operations like addition and multiplication geometrically.

When finding roots, you plot each root as a point. In exercises like this, where roots are equally spaced around a circle, you can visualize them as forming a symmetrical pattern, such as a pentagon if you have five roots.

• Each root is placed at equal angles from its neighbors, dictated by \( \theta_k = \frac{2k\pi}{n} \), where \( n \) is the number of roots.
• The distance from the origin is equal to the magnitude of the root.

This graphical representation offers a great way to see how complex numbers interact and transform with operations.