if|z1|≠|z1+z2|，|z2|≠|z1+z2|，it's easy to check that z1/z2=-(2^|z1|-2^|z1+z2|)/(2^|z2|-2^|z1+z2|)∈R，we amuse that z1=k*z2（k∈R） and we can find that only k=1 satisfy the identity.if|z1|=|z1+z2|=|z2|，we have z1^6=z2^6.

taiat wrote: If $|z_1|\ne |z_1+z_2|, |z_2|\ne |z_1+z_2|$, it's easy to check that $\frac{z_1}{z_2}=-\frac{2^{|z_1|}-2^{|z_1+z_2|}}{2^{|z_2|}-2^{|z_1+z_2|}} \in \mathbb{R}$, we amuse that $z_1=k\cdot z+2 \quad (k\in\mathbb{R})$ and we can find that only $k=1$ satisfies the identity if $|z_1|=|z_1+z_2|=|z_2|$, we have $z_1^6=z_2^6.$