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Question
If a = `(-1 + sqrt(3)"i")/2`, b = `(-1 - sqrt(3)"i")/2` then show that a2 = b and b2 = a
Solution
a = `(-1 + sqrt(3)"i")/2`, b = `(-1 - sqrt(3)"i")/2`
∴ a2 = `((-1 + sqrt(3)"i")/2)^2 = (1 - 2sqrt(3)"i" + 3"i"^2)/4`
= `(1 - 2sqrt(3)"i" + 3(-1))/4` ...[∵ i2 = – 1]
= `(-2 - 2sqrt(3)"i")/4`
= `(-1 - sqrt(3)"i")/2` = b
and b2 = `((-1 - sqrt(3)"i")/2)^2 = (1 + 2sqrt(3)"i" + 3"i"^2)/4`
= `(1 + 2sqrt(3)"i" + 3(-1))/4` . ...[∵ i2 = – 1]
= `(-2 + 2sqrt(3)"i")/4`
= `(-1 + sqrt(3)"i")/2` = a
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