Question
Find the conjugates of the following complex numbers: $\frac{(1+\text{i})(2+\text{i})}{3+\text{i}}$
✓
Answer
Let $\text{Z}=\frac{(1+\text{i})(2+\text{i})}{3+\text{i}}$ $=\frac{2+\text{i}+\text{i}(2+\text{i})}{3+\text{i}}$ $=\frac{2+\text{i}+2\text{i}+\text{i}}{3+\text{i}}$ $=\frac{1+3\text{i}}{3+\text{i}}$ $=\frac{(1+3\text{i})}{(3+\text{i})}\times\frac{(3-\text{i})}{(3-\text{i})}$ $=\frac{3-\text{i}+3\text{i}(3-\text{i})}{3^2+1^2}$ $=\frac{3-\text{i}+9\text{i}+3}{9+1}$ $=\frac{6+8\text{i}}{10}$ $=\frac{2(3+4\text{i})}{10}$ $\Rightarrow\text{z}=\frac{3+4\text{i}}{5}$ Hence $\bar{\text{z}}=\frac{3-4\text{i}}{5}$ $=\frac{3}{5}-\frac{4}{5}\text{i}$
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