Express the mentioned below complex number in the standard form $a + i b$
(xi) $(\frac{1}{1 - 4 i} - \frac{2}{1 + i}) (\frac{3 - 4 i}{5 + i})$

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Solution

$∵ (\frac{1}{1 - 4 i} - \frac{2}{1 + i}) (\frac{3 - 4 i}{5 + i})$

$= [\frac{1 + i - 2 + 8 i}{(1 - 4 i) (1 + i)}] (\frac{3 - 4 i}{5 + i})$

$= [\frac{- 1 + 9 i}{1 + i - 4 i - 4 i^{2}}] (\frac{3 - 4 i}{5 + i})$

$= (\frac{- 1 + 9 i}{5 - 3 i}) (\frac{3 - 4 i}{5 + i})$

$= \frac{- 3 + 4 i + 27 i - 36 i^{2}}{25 + 5 i - 15 i - 3 i^{2}}$

$= \frac{33 + 31 i}{28 - 10 i}$

$= \frac{33 + 31 i}{28 - 10 i} \times \frac{28 + 10 i}{28 + 10 i}$

$= \frac{33 \times 28 + 330 i + 31 \times 28 i + 310 i^{2}}{28^{2} - (10 i)^{2}}$

$= \frac{33 \times 28 + 330 i + 31 \times 28 i + 310 i^{2}}{28^{2} - (10 i)^{2}}$

$= \frac{924 + 330 i + 868 i + 310 \times (- 1)}{784 - 100 i^{2}}$

$= \frac{614 + 1198 i}{884}$

$= \frac{614}{884} + \frac{1198}{884} i$

$= \frac{307}{442} + \frac{599}{442} i$

$∴ (\frac{1}{1 - 4 i} - \frac{2}{1 + i}) (\frac{3 - 4 i}{5 + i}) = \frac{307}{442} + \frac{599}{442} i$

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