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Question
Show that:
`(4 - sqrt5)/(4 + sqrt5) + 2/(5 + sqrt3) + (4 + sqrt5)/(4 - sqrt5) + 2/(5 - sqrt3) = 52/11`
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Solution
`(4 - sqrt5)/(4 + sqrt5) + 2/(5 + sqrt3) + (4 + sqrt5)/(4 - sqrt5) + 2/(5 - sqrt3)`
`= (4 - sqrt5)/(4 + sqrt5) xx (4 - sqrt5)/(4 - sqrt5) + 2/(5 + sqrt3) xx (5 - sqrt3)/(5 - sqrt3) + (4 + sqrt5)/(4 - sqrt5) xx (4 + sqrt5)/(4 + sqrt5) + 2/(5 - sqrt3) xx (5 + sqrt3)/(5 + sqrt3)`
`= (4 - sqrt5)^2/((4)^2 - (sqrt5)^2) + (2(5 - sqrt3))/((5)^2 - (sqrt3)^2) + (4 + sqrt5)^2/((4)^2 - (sqrt5)) + (2(5 + sqrt3))/((5)^2 - (sqrt3)^2)`
`= (16 + 5 - 8sqrt5)/(16 - 5) + (10 - 2sqrt3)/(25 - 3) + (16 + 5 + 8sqrt5)/(16 - 5) + (2(5 + sqrt3))/(25 - 3)`
`= (21 - 8sqrt5)/11 + (10 - 2sqrt3)/22 + (21 + 8sqrt5)/11 + (cancel(2)^1 (5 + sqrt3))/cancel(22)_11`
`= (21 - 8sqrt5)/11 + (cancel(2)^1(5 - sqrt3))/cancel(22)_11 + (21 + 8sqrt5)/11 + (5 + sqrt3)/11`
`= (21 - cancel(8sqrt5) + 5 - cancel(sqrt3) + 21 + cancel(8sqrt5) + 5 + cancel(sqrt3))/11`
`= (21 + 5 + 21 + 5)/11`
`= 52/11`
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