Problem 72 - Algebra and Trigonometry

Question

In Exercises \(69-76,\) find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest tenth. The complex sixth roots of 64

Step-by-Step Solution

Answer

The 6 complex roots are calculated with De Moivre's theorem and converted back into rectangular format. Last step is to round to the nearest tenth if necessary.

Step 1: Identify the roots

Given that it is a sixth roots of 64 problem, it means there are six roots in total for this number.

Step 2: Convert 64 into Complex format

Firstly, convert 64 into its complex polar format, it can be written as 64cis(0°), where 'cis' is shorthand for cos + i sin.

Step 3: Apply De Moivre's theorem

With De Moivre's theorem, to find the \(n\)-th root of a complex number in polar form, \((r cis (θ))\), divide the angle by \(n\) and find all values from \(k = 0\) to \(n - 1\).

Step 4: Calculate the roots

Next, calculate the sixth roots. Here, r = 64 and the roots are \(r^{1/n}\) cis \((θ + 2kπ )/ n\), where \(n\) = 6, \(k\) ranges from 0 to 5.

Step 5: Convert back to Rectangular form

After calculating all 6 roots, convert them back to rectangular form, if necessary.

Step 6: Round to the nearest tenth

If necessary, after calculations and conversions, round all of the complex roots to the nearest tenth

Key Concepts

Rectangular Form

The rectangular form of a complex number is a way to express the number using coordinates on a plane. A complex number is written as \( a + bi \), where:

• \( a \) is the real part.
• \( b \) is the imaginary part, and \( i \) is the imaginary unit with the property \( i^2 = -1 \).

This form is particularly useful when plotting complex numbers on the Cartesian plane, where the real part \( a \) corresponds to the x-axis and the imaginary part \( bi \) corresponds to the y-axis. Visualizing complex numbers this way simplifies operations like addition and subtraction.

De Moivre's Theorem

De Moivre's Theorem is a significant principle in complex number theory, especially useful for finding powers and roots of complex numbers. It states that for any real number \( \theta \) and integer \( n \),\[(r \text{ cis } \theta)^n = r^n \text{ cis } (n\theta)\]where \( \text{cis} \theta = \cos \theta + i \sin \theta \). This theorem greatly simplifies raising complex numbers to powers or extracting roots, as it allows us to multiply the modulus and add the arguments in exponential expressions.
In the context of finding roots, as in the exercise, De Moivre's theorem tells us how to distribute the angle efficiently across the complex plane to find all potential roots.

Polar Coordinates

Polar coordinates for complex numbers offer a different method for representing these numbers. In polar form, a complex number is expressed as \( r \text{ cis } \theta \), where:

• \( r \) is the modulus (or magnitude) of the complex number, representing the distance from the origin to the point in the complex plane.
• \( \theta \) is the argument, indicating the angle formed with the positive real axis.

Using polar coordinates allows for a simple geometric interpretation of multiplication and division of complex numbers. This representation is essential when utilizing De Moivre's Theorem for computations involving powers and roots.

Roots of Unity

"Roots of unity" are special solutions to the equation \( z^n = 1 \), where \( z \) is a complex number. These roots are evenly spaced around the unit circle in the complex plane. For a given \( n \), the \( n \)-th roots of unity are:

• \( 1 \)
• \( e^{2\pi i/n} \)
• \( e^{4\pi i/n} \)
• ... up to ...
• \( e^{2(n-1)\pi i/n} \)

These roots divide the circle into \( n \) equal parts. In practical applications like the exercise, they help in finding all the roots of a complex number when using De Moivre's theorem. The pattern and regularity of these roots allow us to find solutions that otherwise might be cumbersome to calculate directly.