<s>[INST] Find all solutions of $z^5+a^5=0$ The task is as follows:
Find all solutions of $z^5+a^5=0$, where $a$ is a positive real number.

My initial attempt
(which leads nowhere)
My guess is that i'll have to find the 5 5th roots of $-z^5$:
$w_1 = |-z^5|^5(cos(\frac{\theta}{5})+isin(\frac{\theta}{5})) \\
w_1  = |z|(cos(\frac{\theta}{5})+isin(\frac{\theta}{5})$
Then, the other roots are:
$w_2  = |z|(cos(\frac{\theta+2\pi}{5})+isin(\frac{\theta+2\pi}{5}) \\
w_3  = |z|(cos(\frac{\theta+4\pi}{5})+isin(\frac{\theta+4\pi}{5}) \\
w_4  = |z|(cos(\frac{\theta+6\pi}{5})+isin(\frac{\theta+6\pi}{5}) \\
w_5  = |z|(cos(\frac{\theta+8\pi}{5})+isin(\frac{\theta+8\pi}{5})$
but i'm not sure what to do with this new information. Also, I guess $z^5$ have to be a negative real number, since added to $a^5$ its $0$. Again, not sure what to do with this either. 
 [/INST] I figured it out (delayed rubber duck?)
I define $z=(r,\theta)$. Then $z^5=(r^5,5\theta)$. 
Since $z^5$ have to be a real negative number, 
$$ 5\theta=\pi+2\pi n \\ \theta=\frac{\pi+2\pi n}{5}$$ 
And since $$r^5=a^5 \\ r=a$$
Then $z=(a, \frac{\pi+2\pi n}{5})$.
Or, with a more straight forward notation:
$z=a(cos(\frac{\pi+2\pi n}{5})+i sin(\frac{\pi+2\pi n}{5}))$.
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