3. Given a positive integer $n > 2$, find all complex numbers $z$ satisfying $z^n = z^{n-1}$. Find all complex numbers $z$ satisfying $z^n = z^{n-1}$.
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Step 1: The problem asks us to find all complex numbers $z$ that satisfy the equation $z^n = z^{n-1}$ for a given positive integer $n > 2$. Show more…
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Let $z=r(\cos \theta+i \sin \theta)$ be a nonzero complex number, and let $n$ be a positive integer greater than 1. Verify that each of the following $n$ numbers is a solution of the equation $u^{n}=z:$ $$\begin{aligned} &\sqrt[n]{r}\left[\cos \left(\frac{\theta+2 \pi k}{n}\right)+i \sin \left(\frac{\theta+2 \pi k}{n}\right)\right]\\ &k=0,1,2, \ldots, n-1 \end{aligned}$$ where $\sqrt[n]{r}$ denotes the positive real number that, when raised to the $n$ th power, gives $r .$ (Hint: Use De Moivre's Theorem.)
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