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# HELP ASAP

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The equation $z^5 = i$has $5$ solutions. The unique solution in the third quadrant is $re^{i\theta}$, where $r > 0$ and $0 \leq \theta < 2\pi$. What is the ordered pair $(r, \theta)$?​

Jan 17, 2019

#1
+6185
+2

$$z^5 = 1\\ z = \Large e^{i 2\pi \frac{k}{5}}, \normalsize k=0,1,\dots 4$$

$$\text{It might be a bit easier to visualize the quadrants if we write these as degrees}\\ r = (0, 72, 144, 216, 288)^\circ\\ \text{and clearly }180 < 216 < 270\\ 216^\circ \Rightarrow e^{i 2\pi \frac 3 5}\\ r = 1,~\theta = \dfrac{6\pi}{5}$$

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Jan 17, 2019

#1
+6185
+2
$$z^5 = 1\\ z = \Large e^{i 2\pi \frac{k}{5}}, \normalsize k=0,1,\dots 4$$
$$\text{It might be a bit easier to visualize the quadrants if we write these as degrees}\\ r = (0, 72, 144, 216, 288)^\circ\\ \text{and clearly }180 < 216 < 270\\ 216^\circ \Rightarrow e^{i 2\pi \frac 3 5}\\ r = 1,~\theta = \dfrac{6\pi}{5}$$