<s>[INST] $Z^4 = -1$ How do I solve this without a calculator? Basically, the question is to solve $z^8= 1$. I have factored this down to $$(z+1)(z-1)(z^2+1)(z^4+1)=0$$
I have simplified $(z^4+1)$ to  $z =\pm \sqrt{i}$ and online I know that this can be simplified to the following four solutions:
z=  0.7071 + 0.7071 i  
z=  -0.7071 + 0.7071 i  
z=  -0.7071 - 0.7071 i  
z=  0.7071 - 0.7071 i 

How would I get this without a calculator?
Thank you! 
 [/INST] use the Euler's Identity 
$$e^{\pi i}=-1$$
and
$$e^{ix}=\cos x+i\sin x$$
$$z^4=-1$$
$$z^4=e^{\pi i}$$
$$z=e^{\frac{\pi }{4}+\frac{n\pi}{2}}$$
$$z_1=\cos(\frac{\pi }{4}+\frac{\pi}{2})+i\sin(\frac{\pi }{4}+\frac{\pi}{2})$$
$$z_2=\cos(\frac{\pi }{4}+\frac{2\pi}{2})+i\sin(\frac{\pi }{4}+\frac{2\pi}{2})$$
$$z_3=\cos(\frac{\pi }{4}+\frac{3\pi}{2})+i\sin(\frac{\pi }{4}+\frac{3\pi}{2})$$
$$z_4=\cos(\frac{\pi }{4}+\frac{4\pi}{2})+i\sin(\frac{\pi }{4}+\frac{4\pi}{2})$$
</s>