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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:
`(4"i"^8 - 3"i"^9 + 3)/(3"i"^11 - 4"i"^10 - 2)`
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Solution
i8 = (i2)4 = (–1)4 = 1
i9 = i8 × i = (i2)4i = (– 1)4i = i
i11 = i10 × i = (i2)5i = (– 1)5i = – i
i10 = (i2)5 = (– 1)5 = – 1
∴ `(4"i"^8 - 3"i"^9 + 3)/(3"i"^11 - 4"i"^10 - 2) = (4(1) - 3"i" + 3)/(3(-"i") - 4(-1) - 2)`
= `(4 - 3"i" + 3)/(-3"i" + 4 - 2)`
= `(7 - 3"i")/(2 - 3"i")`
= `(7 - 3"i")/(2 - 3"i") xx (2 + 3"i")/(2 + 3"i")`
= `(14 + 21"i" - 6"i" - 9"i"^2)/(4 - 9"i"^2)`
= `(14 + 15"i" + 9)/(4 + 9)` ...[∵ i2 = – 1]
= `(23 + 15"i")/13`
= `23/13 + 15/13"i"`
This is of the form a + bi, where a = `23/13` and b = `15/13`.
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