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Chapter 6: Problem 72

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. \(4(1\ -\ \sqrt{3}i)^3\)

Short Answer

Expert verified
The cube of the given complex number is \(-8\).

Step by step solution

01

Write the Complex Number in Polar Form

A complex number \(a + bi\) can be represented in polar coordinates as \(r(\cos 胃 + i \sin 胃)\), where \(r = \sqrt{a^2 + b^2}\) and \(胃 = arctan(\frac{b}{a})\). Therefore, to write \(1 - \sqrt{3}i\) in polar form, we need to find r and 胃, which are \(\sqrt{1^2 + (-\sqrt{3})^2} = 2\) and \(arctan(\frac{-\sqrt{3}}{1}) = -60掳\) respectively. The polar form of \(1 - \sqrt{3}i\) is, thus, \(2(\cos (-60掳) + i \sin(-60掳))\).
02

Apply DeMoivre's Theorem

DeMoivre's Theorem describes the power of a complex number. It states \((r(\cos 胃 + i \sin 胃))^n = r^n (\cos n胃 + i \sin n胃)\). Replacing \(r = 2\), \(胃 = -60掳\), and \(n=3\), we get: \( (2^3)(\cos 3(-60掳) + i \sin 3(-60掳)) \) which simplifies to \(8( \cos(-180掳) + i \sin(-180掳))\).
03

Convert Back to Rectangular Form

Finally, convert the result back to rectangular form \(a + bi\). Using trigonometric values for the angles, it is found: \[8( \cos(-180掳) + i \sin(-180掳)) \rightarrow 8(-1 + 0i)\] which simplifies to \(-8\).

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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 an extension of the real numbers, and they provide a way to solve equations that do not have solutions within the real numbers alone. They consist of a real part and an imaginary part, often expressed as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The imaginary unit is represented as \(i\), where \(i^2 = -1\). This means that any real number can be considered a complex number with an imaginary part of zero.
  • **Real Part**: The component \(a\) in \(a + bi\).
  • **Imaginary Part**: The component \(b\) in \(a + bi\).
  • **Imaginary Unit (\

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Most popular questions from this chapter

In Exercises 15-32, represent the complex number graphically, and find the trigonometric form of the number. \(12i\)

In Exercises 47-58, perform the operation and leave the result in trigonometric form. \([\frac{5}{3}(\cos\ 120^{\circ} + i\ \sin\ 120^{\circ})][\frac{2}{3}(\cos\ 30^{\circ} + i\ \sin\ 30^{\circ})]\)

TRUE OR FALSE? In Exercises 107 and 108, determine whether the statement is true or false. Justify your answer. Geometrically, the \(n\)th roots of any complex number \(z\) are all equally spaced around the unit circle centered at the origin.

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. \((\cos\ 0 + i\ \sin\ 0)^{20}\)

WORK A force of 45 pounds exerted at an angle of \(30^{\circ}\) above the horizontal is required to slide a table across a floor (see figure). The table is dragged 20 feet. Determine the work done in sliding the table.
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