Find general equation from 2 polar answers

Question

There is $z^n = x+yi$ and $n$ natural $z_1$ and $z_2$ are 2 answers of the equation. $z_1$'s angle is the smallest and $z_2$'s angle is the biggest. $z_1 = \sqrt[4]{2}×e^{i \pi/12}$ $z_2 = \sqrt[4]{2}×e^{i 7 \pi /4 }$

Find $ n,x,y $

How to?

Answer 1

The angle between $z_1$ and $z_2$ is $\frac{\pi}{12}+\frac{\pi}{4}=\frac{\pi}{3}$ radians, If two neighboring roots of a number differ by $\frac16$ of a circle, they must be $6$th roots. Raising $z_1$ to the sixth power, we obtain $z_1^6 = 2^{3/2}e^{\frac{i\pi}{2}}$, or $2^{3/2}i$. Note that the sixth power of $z_2$ is the same.

Thus, it appears that $n=6$, $x=0$, and $y=2\sqrt{2}$

Does that make sense?

Answer 2

Basic approach. Take a look at the difference in arguments (what the original question calls the angle of the root):

$$ \frac{7\pi}{4}-\frac{\pi}{12} = \frac{5\pi}{3} $$

The fundamental difference in arguments between the $n$ roots must be $\frac{2\pi}{n}$ (do you see why?); at the same time, $\frac{5\pi}{3}$ must also be an integer multiple of $\frac{2\pi}{n}$ (again, do you see why?).

At the same time, $\frac{2\pi}{n} > 2\pi-\frac{7\pi}{4} = \frac{\pi}{4}$; otherwise, $z_2$ would not be the root with the highest argument.

What does this permit you to say about the value of $n$? Then raise $z_1$ and $z_2$ to this $n$th power, and see what you obtain. That will be $x+yi$.