Find the conjugates of the following complex numbers: (3-2i)(2+3i… - Vidyadip

Question

Find the conjugates of the following complex numbers:
$\frac{(3-2\text{i})(2+3\text{i})}{(1+2\text{i})(2-\text{i})}$

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Answer

Let $\text{z}=\frac{(3-2\text{i})(2+3\text{i})}{(1+2\text{i})(2-\text{i})}$
$=\frac{3(2+3\text{i})-2\text{i}(2+3\text{i})}{2-\text{i}+2\text{i}(2-\text{i})}$
$=\frac{6+9\text{i}-4\text{i}+6}{2-\text{i}+4\text{i}+2}$
$=\frac{12+5\text{i}}{4+3\text{i}}$
$=\frac{12+5\text{i}}{4+3\text{i}}\times\frac{4-3\text{i}}{4-3\text{i}}$
$=\frac{12(4-3\text{i})+5\text{i}(4-3\text{i})}{4(4-3\text{i})+3\text{i}(4-3\text{i})}$
$=\frac{48-36\text{i}+20\text{i}+15}{16-12\text{i}+12\text{i}+9}$
$=\frac{63-16\text{i}}{16+9}$
$\Rightarrow\text{z}=\frac{63-16\text{i}}{25}$
$\therefore \bar{\text{z}}=\frac{63+16\text{i}}{25}$
$=\frac{63}{25}+=\frac{16}{25}\text{i}$

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