Mathematics, 03.10.2019 12:30 ashiyah74
Find all complex fourth roots of z=-1
Answers: 1
![\bf z=-10000\iff \begin{array}{lcclll} z=&-10000&+0i\\ \end{array}\implies (-10000,0)\\\\ -----------------------------\\\\ r=\sqrt{a^2+b^2}\ \begin{cases} a=10000\\ b=0 \end{cases}\implies r=\sqrt{10000^2+0^2}\implies r=10000 \\\\\\ \textit{now, if you notice the picture below, for those coordinates}\\\\ \textit{the angle will then be }\pi \\\\\\ thus\implies z=10000\left[cos(\pi )+i\ sin(\pi ) \right]](/tpl/images/0286/2824/51094.png)
![\bf \sqrt[n]{z}=\sqrt[n]{r}\left[ cos\left( \frac{\theta+2\pi k}{n} \right) +i\ sin\left( \frac{\theta+2\pi k}{n} \right)\right]\\\\ -----------------------------\\\\ \sqrt[4]{z}=\sqrt[4]{10000}\left[ cos\left( \frac{\pi +2\pi k}{4} \right) +i\ sin\left( \frac{\pi +2\pi k}{4} \right)\right]\\\\ -----------------------------\\\\](/tpl/images/0286/2824/d6261.png)
![\bf \boxed{k=1}\qquad \sqrt[4]{10000}\left[ cos\left( \frac{\pi +2\pi k}{4} \right) +i\ sin\left( \frac{\pi +2\pi k}{4} \right)\right] \\\\\\ 10\left[ cos\left( \frac{3\pi}{4} \right) +i\ sin\left( \frac{3\pi}{4} \right)\right]\implies 10\left(\cfrac{-\sqrt{2}}{2}+i\ \cfrac{\sqrt{2}}{2} \right) \\\\\\ -5\sqrt{2}+i\ 5\sqrt{2}](/tpl/images/0286/2824/0ebc9.png)
![\bf \boxed{k=2}\qquad \sqrt[4]{10000}\left[ cos\left( \frac{\pi +2\pi k}{4} \right) +i\ sin\left( \frac{\pi +2\pi k}{4} \right)\right] \\\\\\ 10\left[ cos\left( \frac{5\pi}{4} \right) +i\ sin\left( \frac{5\pi}{4} \right)\right]\implies 10\left(\cfrac{-\sqrt{2}}{2}+i\ \cfrac{-\sqrt{2}}{2} \right) \\\\\\ -5\sqrt{2}-i\ 5\sqrt{2}](/tpl/images/0286/2824/0f984.png)
![\bf \boxed{k=3}\qquad \sqrt[4]{10000}\left[ cos\left( \frac{\pi +2\pi k}{4} \right) +i\ sin\left( \frac{\pi +2\pi k}{4} \right)\right] \\\\\\ 10\left[ cos\left( \frac{7\pi}{4} \right) +i\ sin\left( \frac{7\pi}{4} \right)\right]\implies 10\left(\cfrac{\sqrt{2}}{2}+i\ \cfrac{-\sqrt{2}}{2} \right) \\\\\\ 5\sqrt{2}-i\ 5\sqrt{2}](/tpl/images/0286/2824/3f1f3.png)
![\bf \boxed{k=4}\qquad \sqrt[4]{10000}\left[ cos\left( \frac{\pi +2\pi k}{4} \right) +i\ sin\left( \frac{\pi +2\pi k}{4} \right)\right] \\\\\\ 10\left[ cos\left( \frac{9\pi}{4} \right) +i\ sin\left( \frac{9\pi}{4} \right)\right]\implies 10\left(\cfrac{\sqrt{2}}{2}+i\ \cfrac{\sqrt{2}}{2} \right) \\\\\\ 5\sqrt{2}+i\ 5\sqrt{2}](/tpl/images/0286/2824/1e0c0.png)
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