Simplify [1 1-2 i +3 1+i ] [3+4 i 2-4 i ] - Vidyadip

Question

Simplify $\left[\frac{1}{1-2 i}+\frac{3}{1+i}\right]\left[\frac{3+4 i}{2-4 i}\right]$

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Answer

${\left[\frac{1}{1-2 i}+\frac{3}{1+i}\right]\left[\frac{3+4 i}{2-4 i}\right]}$
$=\left[\frac{1+i+3-6 i}{(1-2 i)(1+i)}\right]\left[\frac{3+4 i}{2-4 i}\right]$
$=\left[\frac{4-5 i}{1+i-2 i-2 i^2}\right]\left[\frac{3+4 i}{2-4 i}\right]$
$=\frac{(4-5 i)(3+4 i)}{(3-i)(2-4 i)}$
$=\frac{12+16 i-15 i-20 i^2}{6-12 i-2 i+4 i^2}$
$=\frac{12+i+20}{6-14 i-4}=\frac{32+i}{2-14 i}$
$=\frac{(32+i)(2+14 i)}{(2-14 i)(2+14 i)}=\frac{64+448 i+2 i+14 i^2}{4-196 i^2}$
$=\frac{64+450 i-14}{4+196}=\frac{50+450 i}{200}=\frac{50}{200}(1+9 i)$
$=\frac{1}{4}+\frac{9}{4} i$

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