Question

Find all complex fourth roots of z=-10000

Answer 1

`\bf z=-10000\iff \begin{array}{lcclll} z=&-10000&+0i\\ \end{array}\implies (-10000,0)\\\\ -----------------------------\\\\ r=√(a^2+b^2)\ \begin{cases} a=10000\\ b=0 \end{cases}\implies r=√(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]`


`\bf \sqrt[n]{z}=\sqrt[n]{r}\left[ cos\left( (\theta+2\pi k)/(n) \right) +i\ sin\left( (\theta+2\pi k)/(n) \right)\right]\\\\ -----------------------------\\\\ \sqrt[4]{z}=\sqrt[4]{10000}\left[ cos\left( (\pi +2\pi k)/(4) \right) +i\ sin\left( (\pi +2\pi k)/(4) \right)\right]\\\\ -----------------------------\\\\`


`\bf \boxed{k=1}\qquad \sqrt[4]{10000}\left[ cos\left( (\pi +2\pi k)/(4) \right) +i\ sin\left( (\pi +2\pi k)/(4) \right)\right] \\\\\\ 10\left[ cos\left( (3\pi)/(4) \right) +i\ sin\left( (3\pi)/(4) \right)\right]\implies 10\left(\cfrac{-√(2)}{2}+i\ \cfrac{√(2)}{2} \right) \\\\\\ -5√(2)+i\ 5√(2)`


`\bf \boxed{k=2}\qquad \sqrt[4]{10000}\left[ cos\left( (\pi +2\pi k)/(4) \right) +i\ sin\left( (\pi +2\pi k)/(4) \right)\right] \\\\\\ 10\left[ cos\left( (5\pi)/(4) \right) +i\ sin\left( (5\pi)/(4) \right)\right]\implies 10\left(\cfrac{-√(2)}{2}+i\ \cfrac{-√(2)}{2} \right) \\\\\\ -5√(2)-i\ 5√(2)`


`\bf \boxed{k=3}\qquad \sqrt[4]{10000}\left[ cos\left( (\pi +2\pi k)/(4) \right) +i\ sin\left( (\pi +2\pi k)/(4) \right)\right] \\\\\\ 10\left[ cos\left( (7\pi)/(4) \right) +i\ sin\left( (7\pi)/(4) \right)\right]\implies 10\left(\cfrac{√(2)}{2}+i\ \cfrac{-√(2)}{2} \right) \\\\\\ 5√(2)-i\ 5√(2)`


`\bf \boxed{k=4}\qquad \sqrt[4]{10000}\left[ cos\left( (\pi +2\pi k)/(4) \right) +i\ sin\left( (\pi +2\pi k)/(4) \right)\right] \\\\\\ 10\left[ cos\left( (9\pi)/(4) \right) +i\ sin\left( (9\pi)/(4) \right)\right]\implies 10\left(\cfrac{√(2)}{2}+i\ \cfrac{√(2)}{2} \right) \\\\\\ 5√(2)+i\ 5√(2)`

recall that the angle
`\bf \cfrac{9\pi}{4} \iff \cfrac{\pi}{4}`