# Express the following in the form of a + ib, a, b∈R i = −1. State the values of a and b: 2+-34+-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:

(2 + sqrt(-3))/(4 + sqrt(-3))

#### 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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