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Question
Answer the following:
Simplify the following and express in the form a + ib:
`(1 + 2/"i")(3 + 4/"i")(5 + "i")^-1`
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Solution
`(1 + 2/"i")(3 + 4/"i")(5 + "i")^-1`
= `(("i" + 2))/"i".((3"i" + 4))/"i". 1/(5 + "i")`
= `(3"i"^2 + 4"i" + 6"i" + 8)/("i"^2(5 + "i"))`
= `(-3 + 10"i" + 8)/(-1(5 + "i"))` ...[∵ i2 = – 1]
= `((5 + 10"i"))/(-(5 + "i")`
= `((5 + 10"i")(5 - "i"))/(-(5 + "i")(5 - "i"))`
= `(25 - 5"i" + 50"i" - 10"i"^2)/(-(25 - "i"^2)`
= `(25 + 45"i" - 10(-1))/(-[25 - (-1)]`
= `(35 + 45"i")/(-26)`
= `(-35)/26 - 45/26"i"`
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