Question
What is the value of `1/sqrt(12 + sqrt140) +
1/sqrt(8 - sqrt60) - 2/sqrt(10 - sqrt84)` ?Solution
`1/sqrt(12 + sqrt140) + 1/sqrt(8 - sqrt60) - 2/sqrt(10 - sqrt84)` `=1/sqrt((sqrt7)^2 + (sqrt5)^2+ 2sqrt35) + 1/sqrt((sqrt5)^2+ (sqrt3)^2- 2sqrt15) - 2 / sqrt((sqrt7)^2+ (sqrt3)^2- 2sqrt21)` = `1/sqrt((sqrt7+ sqrt5)^2) + 1/sqrt((sqrt5 - sqrt3)^2) - 2/sqrt((sqrt7 - sqrt3)^2)` = `1/(sqrt7+ sqrt5) + 1/(sqrt5 - sqrt3) - 2/(sqrt7 - sqrt3)` = `(sqrt7 - sqrt5)/2 + (sqrt5 + sqrt3)/2 - ((sqrt7+ sqrt3)/2)` = `(sqrt7 - sqrt5 + sqrt5 + sqrt3 - sqrt7 - sqrt3)/2` = 0
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