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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:
`(- sqrt(5) + 2sqrt(-4)) + (1 -sqrt(-9)) + (2 + 3"i")(2 - 3"i")`
Solution
`(- sqrt(5) + 2sqrt(-4)) + (1 -sqrt(-9)) + (2 + 3"i")(2 - 3"i")`
= `(- sqrt(5) + 2sqrt(4).sqrt(-1)) + (1 -sqrt(9).sqrt(-1)) + 2^2 - 9"i"^2`
= `(- sqrt(5) + 2(2)"i") + (1 - 3"i") + 4 - 9"i"^2`
= `-sqrt(5) + 4"i" + 1 - 3"i" + 4 - 9(-1)` ...[∵ i2 = – 1]
= `-sqrt5+"i"+1+4+9`
= `-sqrt5+"i"+14`
= `(14 -sqrt(5)) + "i"`
This is of the form a + bi, where a = `14-sqrt(5)` and b = 1.
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