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Chapter 6: Problem 62

Use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. $$(1-i)^{5}$$

Short Answer

Expert verified
The fifth power of the complex number (1-i) is -4 - 4i.

Step by step solution

01

Convert the complex number to polar form

To convert (1 - i) to polar form, first compute for the modulus \(r = \sqrt{1^2 + (-1)^2} = \sqrt{2}\). The argument \(\theta\) can be calculated using the formula \(\theta = atan2(-1,1) = -\(\frac{\pi}{4}\). So the polar form is \(\sqrt{2} \cdot (cos \(-\(\frac{\pi}{4}\)) + i \cdot sin \(-\(\frac{\pi}{4}\))\)
02

Apply DeMoivre's Theorem

Using DeMoivre's Theorem, we calculate \((1 - i)^5\) = \(\(\sqrt{2}\)^5 \cdot (cos(5 \cdot -\(\frac{\pi}{4}\)) + i \cdot sin(5 \cdot -\(\frac{\pi}{4}\)))\) = \(4\sqrt{2} \cdot (cos \(-\(\frac{5\pi}{4}\)) + i \cdot sin \(-\(\frac{5\pi}{4}\)))\)
03

Convert the result back to rectangular form

The rectangular form is given by \(r \cdot cos \theta + i \cdot r \cdot sin \theta\). Substituting for \(r\) and \(\theta\) from Step 2, we get \(4\sqrt{2} \cdot cos \(-\(\frac{5\pi}{4}\) + i \cdot 4\sqrt{2} \cdot sin \(-\(\frac{5\pi}{4}\)) = -4 - 4i\)

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Most popular questions from this chapter

The components of \(\mathbf{v}=240 \mathbf{i}+300 \mathbf{j}\) represent the respective number of gallons of regular and premium gas sold at a station. The components of \(\mathbf{w}=2.90 \mathbf{i}+3.07 \mathbf{j}\) represent the respective prices per gallon for each kind of gas. Find \(\mathbf{v} \cdot \mathbf{w}\) and describe what the answer means in practical terms.

A force is given by the vector \(\mathbf{F}=3 \mathbf{i}+2 \mathbf{j} .\) The force moves an object along a straight line from the point (4,9) to the point \((10,20) .\) Find the work done if the distance is measured in feet and the force is measured in pounds.

Let $$\mathbf{u}=-\mathbf{i}+\mathbf{j}, \quad \mathbf{v}=3 \mathbf{i}-2 \mathbf{j}, \quad \text { and } \quad \mathbf{w}=-5 \mathbf{j}$$ Find each specified scalar or vector. $$\operatorname{proj}_{\mathbf{u}}(\mathbf{v}-\mathbf{w})$$

Will help you prepare for the material covered in the next section. Refer to Section 2.1 if you need to review the basics of complex numbers. In each exercise, perform the indicated operation and write the result in the standard form \(a+b i\). $$(-1+i \sqrt{3})(-1+i \sqrt{3})(-1+i \sqrt{3})$$

Determine whether v and w are parallel, orthogonal, or neither. $$\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}, \quad \mathbf{w}=6 \mathbf{i}+\frac{18}{5} \mathbf{j}$$
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