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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 + sqrt(-3))/(4 + sqrt(-3))`
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Solution
`(2 + sqrt(-3))/(4 + sqrt(-3)) = (2 + sqrt(3)"i")/(4 + sqrt(3)"i")`
= `((2 + sqrt(3)"i")(4 -sqrt(3)"i"))/((4 + sqrt(3)"i")(4 - sqrt(3)"i"))`
= `(8 - 2sqrt(3)"i" + 4sqrt(3)"i" - 3"i"^2)/(16 - 3"i"^2)`
= `(8 + 2sqrt(3)"i" - 3(-1))/(16 - 3(-1))` ...[∵ i2 = – 1]
= `(11 + 2sqrt(3)"i")/19`
∴ `(2 + sqrt(-3))/(4 + sqrt(-3)) = 11/19 + (2sqrt(3))/19"i"`
∴ a = `11/19` and b = `(2sqrt(3))/19`
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