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Chapter 11: Problem 64

In Exercises \(53-64,\) use DeMoivre's Theorem to find the indicated power of the given complex number. Express your final answers in rectangular form. $$ (1-i)^{8} $$

Short Answer

Expert verified
The power \((1-i)^8\) is 16 in rectangular form.

Step by step solution

01

Convert to Polar Form

First, convert the complex number \((1 - i)\) to polar form. The modulus \(r\) is given by \(r = \sqrt{a^2 + b^2}\), where \(a = 1\) and \(b = -1\). So, \(r = \sqrt{1^2 + (-1)^2} = \sqrt{2}\). The argument \(\theta\) is \(\tan^{-1}\left(\frac{-1}{1}\right) = -\frac{\pi}{4}\). Thus, the polar form is \(\sqrt{2}\left(\cos\left(-\frac{\pi}{4}\right) + i\sin\left(-\frac{\pi}{4}\right)\right)\).
02

Apply DeMoivre's Theorem

According to DeMoivre's Theorem, \((r(\cos \theta + i \sin \theta))^n = r^n(\cos(n\theta) + i\sin(n\theta))\). Here, \(n = 8\), \(r = \sqrt{2}\), and \(\theta = -\frac{\pi}{4}\). Therefore, the expression is \((\sqrt{2})^8(\cos(-2\pi) + i\sin(-2\pi))\).
03

Simplify Exponential and Trigonometric Terms

Compute \((\sqrt{2})^8 = (2^{1/2})^8 = 2^4 = 16\). Evaluate \(\cos(-2\pi)\) and \(\sin(-2\pi)\); we have \(\cos(-2\pi) = 1\) and \(\sin(-2\pi) = 0\). Thus, the expression becomes \(16(1 + i \cdot 0)\).
04

Convert to Rectangular Form

The expression from the previous step is \(16(1 + 0i)\), therefore in rectangular form it is \(16 + 0i \), or simply \(16\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Numbers
Complex numbers are numbers that have two parts: a real part and an imaginary part. 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 as \(i = \sqrt{-1}\), meaning \(i^2 = -1\).
  • Real Part: The real number in the complex number.
  • Imaginary Part: The coefficient of \(i\).
For example, in the complex number \(1 - i\), \(1\) is the real part and \(-1\) is the imaginary part. Complex numbers can be visualized on the complex plane where the x-axis represents the real part and the y-axis represents the imaginary part. This visualization is helpful when working with operations or transformations involving complex numbers.
Polar Form
When dealing with complex numbers, it is often useful to express them in polar form. Polar form represents a complex number based on its modulus and argument rather than its real and imaginary parts.
  • Modulus \(r\): The distance from the origin to the point, calculated as \(r = \sqrt{a^2 + b^2}\).
  • Argument \(\theta\): The angle formed with the positive x-axis, typically computed as \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\).
For the complex number \(1 - i\), the modulus is \(\sqrt{2}\) and the argument is \(-\frac{\pi}{4}\), which allows it to be expressed as \(\sqrt{2}(\cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4}))\). Polar form is particularly useful in calculations involving powers and roots of complex numbers, such as when applying DeMoivre's Theorem.
Rectangular Form
Rectangular form, also known as Cartesian form, is the standard form for writing complex numbers as \(a + bi\). It is most commonly used for basic arithmetic operations such as addition and subtraction. Each complex number is defined by its coordinates on the complex plane.
  • Advantages: Straightforward representation for addition or subtraction.
  • Conversion: Can be converted to polar form for easier multiplication, division, and exponentiation.
In the given example, the complex number was initially converted to polar form and then, after applying DeMoivre's Theorem and simplifying, was returned to rectangular form as \(16 + 0i\), or simply \(16\). Rectangular form clearly shows the real and imaginary components, which is useful in many mathematical and practical applications.
Power of Complex Numbers
DeMoivre's Theorem provides a powerful tool for finding powers of complex numbers. The theorem states that \[(r(\cos \theta + i\sin \theta))^n = r^n(\cos(n\theta) + i\sin(n\theta))\]for any integer \(n\). This means we can easily compute higher powers of a complex number by raising its modulus to the power and multiplying its argument by the power.
  • Easier Computation: DeMoivre's Theorem simplifies exponentiation of complex numbers, especially when they are in polar form.
  • Application: Useful in solving equations involving complex numbers and in fields such as electrical engineering and physics.
In the provided exercise, DeMoivre's theorem enabled the conversion of \((1 - i)^8\) into an expression involving only real numbers: \((\sqrt{2})^8 = 16\), thereby simplifying the computation significantly and leading directly to the rectangular form result of \(16\).

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