解答题
第 76 / 263 题
已知 $z = \frac{2+3i}{1-2i}$，求 $\bar{z}$ 和 $z\bar{z}$。
📖 解析
$z = \frac{(2+3i)(1+2i)}{(1-2i)(1+2i)} = \frac{2+4i+3i+6i^2}{1+4} = \frac{2+7i-6}{5} = \frac{-4+7i}{5} = -\frac{4}{5}+\frac{7}{5}i$，则 $\bar{z} = -\frac{4}{5} - \frac{7}{5}i$。$z\bar{z} = |z|^2 = (-\frac{4}{5})^2 + (\frac{7}{5})^2 = \frac{16}{25}+\frac{49}{25}= \frac{65}{25}= \frac{13}{5}$。