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Question
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(-1)`. State the values of a and b: `((2 + "i"))/((3 - "i")(1 + 2"i"))`
Solution
`((2 + "i"))/((3 - "i")(1 + 2"i")) = (2 + "i")/(3 + 6"i" - "i" - 2"i"^2)`
= `(2 + "i")/(3 + 5"i" - 2(-1)` ...[∵ i2 = – 1]
= `(2 + "i")/(5 + 5"i")`
= `(2 + "i")/(5(1 + "i")) = ((2 + "i")(1 - "i"))/(5(1 + "i
")(1 - "i")`
= `(2 - 2"i" + "i" - "i"^2)/(5(1 - "i"^2)`
= `(2 - "i" - (-1))/(5[1 - (-1)]` ...[∵ i2 = – 1]
= `(3 - "i")/10`
∴ `(2 + "i")/((3 - "i")(1 + 2"i")) = 3/10 - 1/10"i"`
∴ a = `3/10 and "b" = (-1)/10`.
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