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Chapter 7: Problem 123

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$(3-2 i)^{5}$$

Short Answer

Expert verified
The fifth power of the complex number (3-2i) is therefore 122+244i.

Step by step solution

01

Convert Complex number to polar form

The given complex number is \(3 - 2i\). In polar form, \(z = r (\cos \theta + i \sin \theta)\) where \(r = \sqrt{a^{2}+b^{2}}\) and \(\theta = \arctan(\frac{b}{a})\) where \(a\) and \(b\) are the real and imaginary parts of the complex number respectively. Applying the formulas, the modulus \(r = \sqrt{3^{2}+ (-2)^{2}} = \sqrt{13}\) and the argument \(\theta = \arctan(\frac{-2}{3}) = -0.588\). Therefore, \(3 - 2i\) in polar form is \(\sqrt{13} \cos(-0.588) + i \sin(-0.588)\).
02

Apply DeMoivre's Theorem

Applying DeMoivre's Theorem to calculate the fifth power of \(3 - 2i\) gives \((\sqrt{13})^{5} (\cos 5(-0.588) + i \sin 5(-0.588)) = 1487.36 \cos (-2.94) + i \sin (-2.94)\).
03

Convert the result back to rectangular form

Finally, converting back to rectangular form by replacing \(cos\) and \(sin\) with their equivalent rectangular coordinates yield \(1487.36 \times \cos(-2.94) + i (1487.36 \times \sin(-2.94)) = 122 + 244 i\).

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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 include both a real part and an imaginary part. They have a unique representation in the form of \(a + bi\), where \(a\) represents the real component and \(b\) is the coefficient of the imaginary unit \(i\). The imaginary unit \(i\) is defined as the square root of -1, i.e., \(i^2 = -1\).
  • Real Part: The component \(a\), which is a standard real number.
  • Imaginary Part: The component \(bi\), where \(b\) is also real, but multiplied by \(i\).
Complex numbers enable us to solve equations that do not have solutions within the set of real numbers alone, such as \(x^2 + 1 = 0\). Understanding complex numbers is essential in many areas of engineering and physics.
Polar Form
The polar form of a complex number provides an alternative way to represent complex numbers, using a combination of modulus and argument. This representation expresses a complex number \(z = a + bi\) in the form \(r(\cos \theta + i\sin \theta)\), where:
  • \(r\) is the modulus, representing the distance of the complex number from the origin in the complex plane.
  • \(\theta\) is the argument, representing the angle formed with the positive real axis.
Polar form is particularly useful when multiplying and dividing complex numbers, and it is essential for applying DeMoivre's Theorem. This form simplifies the computation of powers and roots of complex numbers by transforming multiplication into addition of angles.
Rectangular Form
The rectangular form of a complex number expresses it in terms of its horizontal and vertical components, commonly \(a + bi\). Unlike polar form, which utilizes angle and distance, rectangular form relates directly to Cartesian coordinates.
  • Horizontal Axis (Real Part): Corresponds to the real part \(a\).
  • Vertical Axis (Imaginary Part): Corresponds to the imaginary part \(bi\).
This form is intuitive for addition and subtraction of complex numbers, as it corresponds to normal vector operations. Transitioning between rectangular and polar forms is often necessary, especially when dealing with powers, such as applying DeMoivre's Theorem.
Modulus and Argument
The modulus and argument are fundamental to describing complex numbers in polar form.
  • Modulus (\(r\)): This is the magnitude or length of the vector representing the complex number on the complex plane. It is found using the formula \(r = \sqrt{a^2 + b^2}\), essentially the distance from the origin to the point \((a, b)\).
  • Argument (\(\theta\)): The angle which the line representing the complex number makes with the positive real axis. It is calculated using \(\theta = \arctan(\frac{b}{a})\). The argument is typically measured in radians.
Understanding both the modulus and argument is critical for converting complex numbers between polar and rectangular forms. They are also key elements when utilizing trigonometric functions to solve complex equations.
Trigonometry
Trigonometry is the branch of mathematics that deals with the relationships between the angles and sides of triangles. It plays a key role in expressing and manipulating complex numbers in polar form, where the trigonometric functions \(\cos\) and \(\sin\) are used to represent the horizontal and vertical components of a complex number, respectively.
  • \(\cos \theta\): The cosine function represents the adjacent side over the hypotenuse in a right triangle, aligning with the real part in complex numbers.
  • \(\sin \theta\): The sine function represents the opposite side over the hypotenuse, aligning with the imaginary part in complex numbers.
Trigonometry simplifies calculations of complex number expressions, especially for raising numbers to powers or finding roots, as is needed in DeMoivre's Theorem. Understanding these concepts allows for efficient manipulation and application of complex numbers in various fields.

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Most popular questions from this chapter

Sketch the graph of all complex numbers \(z\) satisfying the given condition. $$|z|=6$$

Determine whether u and v are orthogonal, parallel, or neither. $$\begin{aligned} &\mathbf{u}=\langle 10,20\rangle\\\ &\mathbf{v}=\langle-18,9\rangle \end{aligned}$$

Find two vectors in opposite directions that are orthogonal to the vector \(\mathbf{u}.\) (There are many correct answers.) $$\mathbf{u}=\langle-7,5\rangle$$

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$(1+i)^{3}$$

Find \(\mathbf{u} \cdot \mathbf{v},\) where \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{v}.\) $$\|\mathbf{u}\|=100,\|\mathbf{v}\|=250, \theta=\frac{\pi}{6}$$
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