Prove that $Z=\frac {\left( r_{1}r_{2}+z_{1}z_{2}\right) ^{n}}{\left( r_{1}+z_{1}\right) \left( r_{2}+z_{2}\right) }\in \mathbb{R} $

Question

We consider the complex numbers $ z_{1},z_{2}$ such as $|z_{i}|=r_{i}$.

Prove that $$Z=\frac {\left( r_{1}r_{2}+z_{1}z_{2}\right) ^{n}}{\left( r_{1}+z_{1}\right) \left( r_{2}+z_{2}\right) }\in \mathbb{R} $$

Answer 1

This is not true, let:

$$z_1=z_2=1+i\tag1$$

Then we know:

$$r_1=r_2=\left|1+i\right|=\sqrt{2}\tag2$$

So, when letting $\text{n}=4$:

$$\frac{\left(\sqrt{2}\cdot\sqrt{2}+\left(1+i\right)\cdot\left(1+i\right)\right)^4}{\left(\sqrt{2}+1+i\right)\cdot\left(\sqrt{2}+1+i\right)}=16-16\sqrt{2}+16\cdot\left(\sqrt{2}-1\right)\cdot i\tag3$$

Answer 2

It is not true: take $z_1=1,z_2=i$ and $n=2$. Then $Z=\frac{1}{2}(1+i)$.