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Chapter 6: Problem 84

Solve each equation in the complex number system. Express solutions in polar and rectangular form. $$x^{5}-32 i=0$$

Short Answer

Expert verified
The solutions in polar form are \(2(cos ({\frac{\pi}{10}}+ k* {\frac{2\pi}{5}}) + i sin ({\frac{\pi}{10}}+ k* {\frac{2\pi}{5}}))\) and in rectangular form as \(2cos ({\frac{\pi}{10}}+ k* {\frac{2\pi}{5}}) + 2i sin ({\frac{\pi}{10}}+ k* {\frac{2\pi}{5}})\) where \(k={0,1,2,3,4}\)

Step by step solution

01

Re-write the equation and express \(32i\) in polar form

The given equation can be rewritten as \(x^{5}=32i\). To better work with the value \(32i\), it's helpful to convert it into polar form. A complex number \(z=x+yi\) can be written in polar form as \(r(cos \theta + i sin \theta)\) where \(r=\sqrt{x^{2} + y^{2}}\) and \(\theta = atan2(y,x)\). This gives us, \(32i\) as \(32(cos {\frac{\pi}{2}} + isin {\frac{\pi}{2}})\) in polar form.
02

Calculate the fifth root

Next, we take the fifth root of both sides of the equation. The fifth root of a number in polar form is found by taking the fifth root of \(r\) and dividing the angle \(\theta\) by 5. The fifth root of \(32\) is \(2\) and \({\frac{\pi}{2}}/5={\frac{\pi}{10}}\). However, there are 5 fifth roots to any complex number, so we need to add a multiple of \(2\pi\) to represent all the possible angles. So, the solution can be generally represented as \(2(cos ({\frac{\pi}{10}}+ k* {\frac{2\pi}{5}}) + i sin ({\frac{\pi}{10}}+ k* {\frac{2\pi}{5}}))\) where \(k={0,1,2,3,4}\).
03

Convert back to rectangular form

The final step involves converting these solutions back into rectangular form. This is done by expanding the polar form using Euler鈥檚 formula \(e^{i\theta}= cos \theta + i sin \theta\). This gives the solutions in rectangular form as \(2cos ({\frac{\pi}{10}}+ k* {\frac{2\pi}{5}}) + 2i sin ({\frac{\pi}{10}}+ k* {\frac{2\pi}{5}})\) where \(k={0,1,2,3,4}\).

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Most popular questions from this chapter

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