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Chapter 6: Problem 75

Find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest tenth. The complex fourth roots of \(1+i\)

Short Answer

Expert verified
By following the steps, we can find that the fourth roots of \(1 + i\) in rectangular form are: \(1\), \(i\), \(-1\), \(-i\).

Step by step solution

01

Convert to Polar Form

To start with, write the complex number \(1 + i\) into polar form. This involves finding the magnitude (r) and the argument (\(\theta\)). The magnitude \(r\) is given by \(\sqrt{a^2 + b^2}\), where \(a\) and \(b\) are the real and imaginary parts of the complex number. Meanwhile, \(\theta\) is given by \(\arctan(\dfrac{b}{a})\). Applying these to the complex number \(1 + i\), where \(a = 1\) and \(b = 1\), compute \(r\) and \(\theta\).
02

Apply De Moivre's Theorem

De Moivre's Theorem says that if you have a complex number in polar form, the n-th roots of that number can be found by dividing the argument by n and having the root values spaced equally around the unit circle. If the formula for the complex number in polar form is written as \(r(\cos\theta + i\sin\theta)\), then the k-th n-th root is given by \(\sqrt[n]{r}(\cos(\frac{\theta+2k\pi}{n}) + i\sin(\frac{\theta+2k\pi}{n}))\), for \(k = 0\) to \(n-1\). Apply this now with \(n = 4\).
03

Convert Back to Rectangular Form

Once you've got all the roots in polar form, convert them back into rectangular form. Remember that for a complex number written in the form \(r(\cos\theta + i\sin\theta)\), its rectangular form equivalent is \(r\cos\theta + ri\sin\theta\).

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