[tex]\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][/tex]
[tex]\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]\\\\
-----------------------------\\\\[/tex]
[tex]\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}[/tex]
[tex]\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}[/tex]
[tex]\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}[/tex]
[tex]\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}[/tex]
recall that the angle [tex]\bf \cfrac{9\pi}{4} \iff \cfrac{\pi}{4}[/tex]
