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Question
In the following, find the value of a and b:
`(7 + sqrt(5))/(7 - sqrt(5)) - (7 - sqrt(5))/(7 + sqrt(5)) = "a" + "b"sqrt(5)`
Solution
`(7 + sqrt(5))/(7 - sqrt(5)) - (7 - sqrt(5))/(7 + sqrt(5)`
= `(7 + sqrt(5))/(7 - sqrt(5)) xx (7 + sqrt(5))/(7 + sqrt(5)) - (7 - sqrt(5))/(7 + sqrt(5)) xx (7 - sqrt(5))/(7 - sqrt(5))`
= `((7 + sqrt(5))^2)/(7^2 - (sqrt(5))^2) - (7 - sqrt(5))^2/(7 ^2 - (sqrt(5))^2`
= `(7^2 + 2 xx 7 xx sqrt(5) + (sqrt(5))^2)/(49 - 5) - (7^2 - 2 xx 7 xx sqrt(5) + (sqrt(5))^2)/(49 - 5)`
= `(49 + 14sqrt(5) + 5)/(44) - (49 - 14sqrt(5) + 5)/(44)`
= `(54 + 14sqrt(5))/(44) - (54 - 14sqrt(5))/(44)`
= `(2(27 + 7sqrt(5)))/(44) - (2(22 - 7sqrt(5)))/(44)`
= `(27 + 7sqrt(5))/(22) - (27 - 7sqrt(5))/(22)`
= `(27)/(22) + (7sqrt(5))/(22) - (27)/(22) + (7sqrt(5))/(22)`
= `(14sqrt(5))/(22)`
= `(7sqrt(5))/(11)`
= `0 + (7sqrt(5))/(11)`
= `"a" + "b"sqrt(5)`
Hence, a = 0 and b = `(7)/(11)`.
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