Express the following complex numbers in the standard form a + ib… - Vidyadip

Question

Express the following complex numbers in the standard form a + ib:
$\frac{3-4\text{i}}{(4-2\text{i})(1+\text{i})}$

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Answer

$\frac{3-4\text{i}}{(4-2\text{i})(1+\text{i})}=\frac{3-4\text{i}}{4(1+\text{i})-2\text{i}(1+\text{i})}$
$=\frac{3-4\text{i}}{4+4\text{i}-2\text{i}+2}$
$=\frac{3-4\text{i}}{6+2\text{i}}$
$=\frac{3-4\text{i}}{6+2\text{i}}\times\frac{6-2\text{i}}{6-2\text{i}}$
$=\frac{3(6-2\text{i})-4\text{i}(6-2\text{i})}{6^2+2^2}$
$=\frac{18-6\text{i}-24\text{i}-8}{36+4}$
$=\frac{10(1-3\text{i})}{40}$
$=\frac{1-3\text{i}}{4}$
$=\frac{1}{4}-\frac{3}{4}\text{i}$
$\therefore \ \frac{3-4\text{i}}{(4-2\text{i})(1+\text{i})}=\frac{1}{4}-\frac{3}{4}\text{i}$

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