106新竹高中

計算五的第二小題

\(\displaystyle \sum_{k=1}^8 z_k{}^7 = \sum_{k=1}^8 \frac{z_k{}^8}{z_k} = \left(-4+5i\right)\cdot\sum_{k=1}^8 \frac{1}{z_k} = \left(-4+5i\right)\cdot \frac{\left(z_1,z_2,\cdots z_8\mbox{任取7個乘積之和}\right)}{z_1 z_2\cdots z_8}=0\)