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Chapter 6: Problem 83

Solve each equation in the complex number system. Express solutions in polar and rectangular form. $$x^{4}+16 i=0$$

Short Answer

Expert verified
The solution in polar form are \(2[\cos(-\frac{\pi}{8})+i\sin(-\frac{\pi}{8})]\), \(2[\cos(\frac{5*\pi}{8})+i\sin(\frac{5*\pi}{8})]\), \(2[\cos(\frac{9*\pi}{8})+i\sin(\frac{9*\pi}{8})]\) and \(2[\cos(\frac{13*\pi}{8})+i\sin(\frac{13*\pi}{8})]\). The solution in rectangular form should be calculated by the given polar forms.

Step by step solution

01

Rearrange the equation

Rearrange the equation to isolate \(x^{4}\) on one side: \[x^{4}=-16 i\]
02

Calculate the fourth root

Calculate the fourth root on both sides. \[x=\sqrt[4]{-16 i}\]
03

Convert the number into polar form

Convert \(-16i\) into polar form. The radius r is \(\sqrt{16^2}=16\) and the argument \(\theta\) is \(\theta = arctan(\frac{-16}{0})\), which equals \(-\frac{\pi}{2}\). So, our number in polar form is \(16(\cos(-\frac{\pi}{2})+i\sin(-\frac{\pi}{2})\)
04

Calculate fourth root in polar form

Now to calculate fourth root in polar form remember that \(\sqrt[4]{z}=\sqrt[4]{r}[\cos(\frac{\theta}{4})+i\sin(\frac{\theta}{4})]\). So, you end up with four solutions all with radius \(\sqrt[4]{16} = 2\), but the arguments are different \(\frac{-\pi/2}{4}=\frac{-\pi/8}\), \(\frac{3*\pi-\pi/2}{4}=\frac{5*\pi/8}\), \(\frac{5*\pi-\pi/2}{4}=\frac{9*\pi/8}\), \(\frac{7*\pi-\pi/2}{4}=\frac{13*\pi/8}\). Plug these arguments into the equation to get your solutions
05

Convert solutions into rectangular form

Calculate the rectangular form for each root by using \(z=r[\cos(\theta)+i\sin(\theta)]\). After doing this you will get four results which are your roots in rectangular form

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