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Question
Answer the following:
Show that z = `((-1 + sqrt(-3))/2)^3` is a rational number
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Solution
`((-1 + sqrt(-3))/2)^3 = ((-1 + sqrt(3 xx - 1))/2)^3`
= `((-1 + sqrt(3)"i")/2)^3`
= `(-1 + sqrt(3)"i")^3/8`
= `1/8[(-1)^3 + 3(-1)^2 sqrt(3)"i" + 3(-1)(sqrt(3)"i")^2 + (sqrt(3)"i")^3]`
= `1/8[-1 + 3sqrt(3)"i" - 3 xx 3"i"^2 + 3sqrt(3)"i"^3]`
= `1/8[-1 + 3sqrt(3)"i" + 9 - 3sqrt(3)"i"]` ...[∵ i2 = – 1, i3 = – i]
= `1/8(8)`
= 1, which is a rational number.
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