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Question
Show that `((sqrt(7) + "i"sqrt(3))/(sqrt(7) - "i"sqrt(3)) + (sqrt(7) - "i"sqrt(3))/(sqrt(7) + "i"sqrt(3)))` is real
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Solution
`((sqrt(7) + sqrt(3)"i")/(sqrt(7) - sqrt(3)"i") + (sqrt(7) - sqrt(3)"i")/(sqrt(7) + sqrt(3)"i"))`
= `[((sqrt(7) + sqrt(3)"i")^2 + (sqrt(7) - sqrt(3)"i")^2)/((sqrt(7) -sqrt(3)"i")(sqrt(7) + sqrt(3)"i"))]`
= `(7 + 2sqrt(21)"i" + 3"i"^2 + 7 - 2sqrt(21)"i" + 3"i"^2)/(7 - 3"i"^2)`
= `(7 + 3(-1) + 7 + 3(-1))/(7 - 3(-1)` ...[∵ i2 = – 1]
= `(7 - 3 + 7 - 3)/(7 + 3)`
= `8/10`
= `4/5`
which is a real number
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