Finding all roots of equations which solutions are different than $z=0$ $$2\overline z = z^7 \, , \,\,32\overline z = z^7 \,,\,\, 128\overline z + z^7=0 $$
Let's say that $z=r\\$ then $2r=r^7$ then $2=r^6$ which gives us $r=\sqrt[6]{2}$.
Now we take (argument) $$\text{arg}\, z=\alpha\,, \,\,\text{arg}\,\overline z=-\alpha\, , \,\,\text{arg}\,2\overline z=-\alpha$$ so $$-\alpha=7 \alpha \, , \,\,8\alpha=0\,,\,\,8\alpha=2k\pi\,,\,\, \alpha=k\frac {\pi}{4}.$$
Am I doing this the correct way? Can someone explain what am I supposed to do next? How can I solve other equations?
 A: Let me try to solve $$z^n = t\bar{z}^m$$ ($m$ and $n$ are positive integers).  If $t=0$, the only solution is $z=0$.  Suppose that $t\ne 0$.  Clearly, $z=0$ is a still solution.  We are seeking all solutions $z\neq 0$.
If $m=n$, we have
$$\left(\frac{z}{\bar{z}}\right)^n=t.$$
Taking modulus we get $|t|=1$ is the only way to have a non-zero soln.  If $t=e^{\theta i}$ for some $\theta\in [0,2\pi)$.  That is, $$\frac{z}{\bar{z}}=e^{\theta i+\frac{2\pi k i}{n}}$$
for some $k=0,1,\dots,n-1$.  That is, $$z=r e^{\frac{\theta i}{2}+\frac{\pi k i}{n}}$$ for some $k=0,1,\dots,n-1$ and $r\in\Bbb{R}$ s.t. $r\ne 0$.
If $m\neq n$, we have $$|z|^{n-m}=\left|\frac{z^n}{\bar{z}^m}\right|=|t|.$$
So, $|z|=|t|^{\frac{1}{n-m}}$.  That is,
$$z=|t|^{\frac{1}{n-m}}e^{\phi i}$$
for some $\phi\in[0,2\pi)$.  Plugging this into $z^n=t\bar{z}^m$, we have
$$|t|^{\frac{n}{n-m}}e^{n\phi i}=t|t|^{\frac{m}{n-m}}e^{-m\phi i}.$$
Let $t=|t|e^{\theta i}$ for some $\theta\in[0,2\pi)$.  We get
$$e^{(m+n)\phi i}=e^{\theta i}.$$
So, $\phi = \frac{\theta}{m+n}+\frac{2k\pi}{m+n}$ for $k=0,1,\dots,m+n-1$.  I then get
$$z=|t|^{\frac{1}{n-m}}e^{\frac{\theta i}{m+n}+\frac{2k\pi i}{m+n}}$$
for $k=0,1,\dots,m+n-1$.
A: HINT
Multiplying by $z$ we obtain
$$ 2\overline zz = z^8 \iff z^8=2|z|^2 \implies |z|^6=2 \implies |z|=2^\frac16$$
and then
$$z^8=2^\frac43 \implies z=\ldots$$
A: All equations are of the form
$$a\overline z=z^7$$
with $a$ real.
Switching to polar coordinates,
$$|a|r=r^7$$ and $$\theta_a-\theta_r=7\theta_r+2k\pi$$ where $\theta_a$ is $0$ or $\pi$.
Then
$$z=\sqrt[6]{|a|}\text{ cis}\frac{\theta_a+2k\pi}8.$$
