# Express the following in the form of a + ib, a, b∈R i = −1. State the values of a and b: (1 + i)−3 - Mathematics and Statistics

Sum

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

#### 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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