Problem 39 - A Graphical Approach to Precalculus with Limits

Question

Find all indicated roots and express them in rectangular form. Check your results with a calculator. The fourth roots of 81.

Step-by-Step Solution

Answer

The fourth roots of 81 are 3, 3i, -3, and -3i.

Step 1: Understanding the Problem

We need to find the fourth roots of 81. This means we are looking for all the complex numbers which, when raised to the power of 4, result in 81.

Step 2: Express 81 in Exponential Form

81 is a real number and can be expressed as a power of another number. Notice that 81 can be written as \(3^4\), since \(3 \times 3 \times 3 \times 3 = 81\). Thus, \(81 = 3^4\).

Step 3: Use De Moivre's Theorem

De Moivre’s Theorem assists in finding the roots of complex numbers. It states that for a complex number \(r(cos \theta + i \sin \theta)\), its \(n\)-th roots are given by \[\sqrt[n]{r} \left(\cos \left(\frac{\theta + 2k\pi}{n}\right) + i \sin \left(\frac{\theta + 2k\pi}{n}\right)\right)\] for \(k = 0, 1, 2, ..., n-1\).

Step 4: Find 81 in Complex Plane

Since 81 is positive and on the real axis, it can be seen as 81 + 0i. In polar form, it is written as \(81e^{i0}\). With our previous calculation, this is \((3^4)e^{i0}\).

Step 5: Calculate the Fourth Root's Magnitude

The fourth root of the magnitude (dr-real number part) is \(\sqrt[4]{3^4} = 3\). So, each fourth root has a magnitude of 3.

Step 6: Determine the Angles for the Roots

Given \(\theta = 0\), and using De Moivre’s Theorem with \(n = 4\), calculate the angle as \(\frac{0 + 2k\pi}{4}\) for \(k = 0, 1, 2, 3\). This results in angles of 0, \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\).

Step 7: Convert Roots to Rectangular Form

Compute each fourth root using the angles found:- For \(k=0\): \(3(\cos 0 + i \sin 0) = 3 + 0i\)- For \(k=1\): \(3(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2}) = 0 + 3i\)- For \(k=2\): \(3(\cos \pi + i \sin \pi) = -3 + 0i\)- For \(k=3\): \(3(\cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2}) = 0 - 3i\).

Step 8: Verify with a Calculator

Input each obtained root into a calculator and raise it to the fourth power to check they yield 81. Each calculated root, when raised to the fourth power, indeed gives 81.

Key Concepts

De Moivre's Theorem

De Moivre's Theorem is a powerful tool in complex number mathematics. It provides a straightforward method to calculate the powers and roots of complex numbers.
This theorem is particularly useful when working with polar coordinates, where a complex number is expressed in terms of a magnitude and an angle.
The theorem states that for any complex number written in polar form as \( r(\cos \theta + i \sin \theta) \), the \( n \)-th roots of this number are given by:

• \( \sqrt[n]{r} \left( \cos \left( \frac{\theta + 2k\pi}{n} \right) + i \sin \left( \frac{\theta + 2k\pi}{n} \right) \right) \)

where \( k \) is a whole number ranging from 0 to \( n-1 \).

In the context of finding fourth roots, as in the exercise provided, we set \( n = 4 \) and calculate the different angles based on \( k = 0, 1, 2, \) and \( 3 \). These angles correspond to different positions on the complex plane, ensuring we locate all the distinct roots of a complex number.

Rectangular Form

The rectangular form of a complex number expresses it in terms of its real and imaginary components.
This is usually written in the form \( a + bi \), where \( a \) is the real part and \( bi \) the imaginary part.
Converting from polar to rectangular form involves using trigonometric functions.
Given a complex number in polar form \( r(\cos \theta + i \sin \theta) \), its rectangular form is found by:

• Using \( x = r \cos \theta \), which is the real part
• Using \( y = r \sin \theta \), which is the imaginary part

These calculations provide each complex root in a format that's perhaps more familiar and easy to work with for many students.
In the exercise, we saw the transformation of the fourth roots into rectangular form as \( 3 + 0i, 0 + 3i, -3 + 0i, \) and \( 0 - 3i \). Each of these corresponds to a specific angle and magnitude in the complex plane.

Polar Coordinates

Polar coordinates provide a means of expressing complex numbers by using a magnitude and an angle.
This form is especially handy when performing multiplications or finding roots and powers of complex numbers.
A point in the complex plane is defined in polar coordinates by two values:

• \( r \), the distance from the origin, which is the magnitude of the complex number
• \( \theta \), the angle made with the positive x-axis, which is known as the argument

This form is articulated as \( r(\cos \theta + i \sin \theta) \), or succinctly as \( re^{i\theta} \).
In the original exercise, expressing 81 in polar form yielded \( 81e^{i0} \).
This shows it as a point 81 units along the positive real axis.
From here, the fourth roots were derived using De Moivre's Theorem by dividing the angle 0 into four equal parts. Each part represented a different root's angular position, translating back into rectangular components.