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Question
Express the following complex number in the standard form a + i b:
\[\left( \frac{1}{1 - 4i} - \frac{2}{1 + i} \right)\left( \frac{3 - 4i}{5 + i} \right)\]
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Solution
`(1/(1 - 4i) - 2/(1 + i))((3 - 4i)/(5 + i))`
= `(1 + i - 2(1 - 4i))/((1 - 4i)(1 + i)) xx (3 - 4i)/(5 + i)`
= `(1 + i - 2 + 8i)/(1(1 + i)-4i(1 + i))xx (3 - 4i)/(5 + i)`
= `(-1+9i)/((1 + i - 4i + 4)) xx (3 - 4i)/(5 + i)`
= `(-1 + 9i)/((1 + i - 4i + 4)) xx (3 - 4i)/(5 + i)`
= `(-1(3 - 4i) + 9i(3 - 4i))/((5 - 3i)(5 + i))`
= `(-3 + 4i + 27i + 36)/(5(5 + i)-3i(5 + i))`
= `(33 + 31j)/(25 + 5i - 15i + 3)`
= `(33 + 31j)/(28 - 10i)`
= `(33 + 31j)/(28 - 10i) xx ((28 + 10i))/(28 + 10i)`
= `(33 xx 28 + 33 xx 10i + 31i xx 28 + 31i xx 10i)/(28^2 + 10^2)`
= `(924 + 330i + 868i - 310)/(784 + 100)`
= `(614 + 1198i)/(884)`
= `614/884 + (1198)/884 i`
= `307/442 + 599/442 i`
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