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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:
(1 + i)−3
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Solution
(1 + i)−3 = `1/((1 + "i")^3`
= `1/(1 + 3"i" + 3"i"^2 + "i"^3)`
= `1/(1 + 3"i" - 3 - "i")` ...[∵ i2 = – 1, i3 = – i]
= `1/(-2 + 2"i")`
= `1/(-2 + 2"i") xx (-2 - 2"i")/(-2 - 2"i")`
= `(-2 - 2"i")/(4 - 4"i"^2)`
= `(-2 - 2"i")/(4 + 4)` ...[∵ i2 = – 1]
= `(-2 -2"i")/8`
= `-1/4 - 1/4 "i"`
This is of the form a + bi, where a = `-1/4` and b = `-1/4`
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