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Question
Reduce `(1/(1-4i) - 2/(1+i))((3-4i)/(5+i))` to the standard form.
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Solution
`(1/(1-4i) - 2/(1+i))((3-4i)/(5+i)) = [((1 + i) - 2(1 + 4i))/((1 - 4i)(1 + i)]] [ (3 - 4i)/(5 +i)]`
= `[(1 + i - 2 + 8i)/(1 + i - 4i - 4i^2)][(3 - 4i)/(5 +i)] = [(- 1 + 9i)/(5 - 3i)] [(3 - 4i)/(5 + i)]`
= `[( - 3 + 4i + 27i - 36i^2)/(25 + 5i - 15i - 3i^2)] = (33 + 31i)/(28 - 10i) =(33 + 31i)/(2(14 - 5i)`
= `(33 + 31i )/(2(14 - 5i)) xx (14 + 5i)/(14 + 5i)`
= `(462 + 165i + 434i + 155i^2)/(2[(14)^2 - (5i)^2]] xx (307 + 599i)/(2(196 - 25i^2)`
= `(307 + 599i)/(2(221)) = (307 + 599i)/442 = 307/442 + (599i)/442`
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