Problem 12 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-12: Use De Moivre's theorem to change the given complex number to the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ (-2-2 i)^{10} $$

Step-by-Step Solution

Answer

(-2 - 2i)^{10} = -32768

Step 1: Express the complex number in polar form

The complex number \(-2 - 2i\) can be expressed in polar form as \(r \text{cis} \theta\). Here, \(r\) is the magnitude and \(\theta\) is the argument. The magnitude \(r\) is computed as \(r = \sqrt{(-2)^2 + (-2)^2} = \sqrt{8} = 2\sqrt{2}\). The argument \(\theta\) is \(\tan^{-1}(-1)\), which gives us \(\theta = -\frac{3\pi}{4}\) or equivalent in the correct quadrant.

Step 2: Apply De Moivre's Theorem

According to De Moivre's Theorem, the polar form of a complex number raised to a power \(n\) is \(r^n \text{cis}(n\theta)\). Here, \(r = 2\sqrt{2}\), \(\theta = -\frac{3\pi}{4}\), and \(n = 10\).The theorem gives us \[(r \text{cis} \theta)^{10} = (2\sqrt{2})^{10} \text{cis} \left(10\left(-\frac{3\pi}{4}\right)\right).\]

Step 3: Calculate the magnitude

Calculate \((2\sqrt{2})^{10}\). Since \(2\sqrt{2} = 2^{1.5}\), \((2\sqrt{2})^{10} = (2^{1.5})^{10} = 2^{15}\).Therefore, the magnitude is \(2^{15} = 32768\).

Step 4: Calculate the angle

Calculate the angle \(10\left(-\frac{3\pi}{4}\right)\). This simplifies to \(-\frac{30\pi}{4} = -\frac{15\pi}{2}\), which is equivalent to \(-7\pi\).

Step 5: Convert back to rectangular form

Convert from polar to rectangular form using \(r(\cos \theta + i \sin \theta)\), where \(r = 32768\) and \(\theta = -7\pi\). The cos and sin of \(-7\pi\) are \(-1\) and \(0\), respectively. Thus, \(32768(-1 + 0i) = -32768\).

Step 6: Final Check

Verify our computation to confirm that \((r\text{cis}\theta)^{10} = -32768\). The magnitude is correct as \(2^{15} = 32768\), and the angle processing aligns correctly with the trigonometric values at multiples of \(\pi\). Thus, the complex number in rectangular form is correct.

Key Concepts

Complex Numbers

Complex numbers are like super numbers that include both real and imaginary parts. They are often written in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The imaginary unit \(i\) is defined such that \(i^2 = -1\). This special type of number can be used to solve equations that normal real numbers can't solve. For instance, the equation \(x^2 + 1 = 0\) has no real solution, but has a solution in the complex numbers: \(x = i\).
Complex numbers are used all over the place in engineering, physics, and even computer science. They help in representing things like electrical circuits, signal processing, and fluid dynamics. Understanding how they work can be a real power-up for anyone diving into the world of math and science.

Polar Form

The polar form of a complex number is a way to express it using the magnitude (or length) and the angle, instead of using real and imaginary parts. Think of it like describing a point in a circle. You say how far out you go (that's the magnitude) and what angle to turn (that's the argument).

Converting a complex number like \(-2 - 2i\) into polar form involves:

• Calculating the magnitude, \(r\), which is the distance from the origin in the Argand plane to the point, using the formula \(r = \sqrt{a^2 + b^2}\).
• Finding the argument, \(\theta\), which is the angle from the positive x-axis, using the arctangent function: \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\).

For \(-2 - 2i\), the magnitude is \(2\sqrt{2}\), and the argument in radians is \(-\frac{3\pi}{4}\). Once in polar form, operations like multiplication and division of complex numbers become much simpler.

Rectangular Form

Rectangular form is another way of representing complex numbers, which you've probably seen as \(a + bi\). Each part has a specific role: the real part, \(a\), tells how far to move along the real axis, and the imaginary part, \(b\), tells how far to move along the imaginary axis on the complex plane.

This form is very useful for straightforward addition and subtraction of complex numbers. For example, to add two complex numbers, \((a_1 + b_1i)\) and \((a_2 + b_2i)\), you simply add their real parts and their imaginary parts individually:

• Real part: \(a_1 + a_2\)
• Imaginary part: \(b_1 + b_2\)

This simplicity makes rectangular form perfect for solving many real-world problems involving complex numbers.
In the provided exercise, once calculated, the rectangular form of the complex number is \(-32768 + 0i\), which simplifies to just \(-32768\).

Magnitude and Argument

The magnitude and argument are two components of the polar form of a complex number. They allow us to understand the position of the complex number on the complex plane by thinking of it as a vector.

The magnitude, sometimes called the modulus, is the "length" of the complex number from the origin (0,0) to the point \((a, b)\). It's found using the Pythagorean theorem: \(r = \sqrt{a^2 + b^2}\). In our exercise, this was computed as \(2\sqrt{2}\) for \(-2 - 2i\).
Meanwhile, the argument is the direction of the vector, or the angle it forms with the positive x-axis. This can sometimes be tricky due to which quadrant the complex number is located in, but we compute it using \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\). For \(-2 - 2i\), the argument is \(-\frac{3\pi}{4}\).
These two components are crucial when converting between polar and rectangular forms, especially in calculations like multiplication and exponentiation of complex numbers using De Moivre's Theorem.