Show that: `1/(3-sqrt(8))-1/(sqrt(8)-sqrt(7))+1/(sqrt(7)-sqrt(6))-1/(sqrt(6)-sqrt(5))+1/(sqrt(5)-2)=5`

Explanation

We hve.
`(1)/(3- sqrt(8)) - (1)/(sqrt(8) - sqrt(7)) + (1)/(sqrt(7) - sqrt(6)) - (1)/(sqrt(6) - sqrt(5)) + (1)/(sqrt(5) -2)`
` = (1)/((3 - sqrt(8))) - ((3 + sqrt(8)))/((3+sqrt(8))) - (1)/((sqrt(8) - sqrt(7))) xx((sqrt(8) + sqrt(7)))/((sqrt(8) + sqrt(7)))+ (1)/(sqrt(7) -sqrt(6)) xx ((sqrt(7) + sqrt(6)))/((sqrt(7) + sqrt(6)))-(1)/((sqrt(6) - sqrt(5)))xx((sqrt(6) + sqrt(5)))/((sqrt(6) + sqrt(5))) + (1)/((sqrt(5) - 2)) xx ((sqrt(5) + 2sqrt(2)))/((sqrt(5) + sqrt(2)))`
`=((3 + sqrt(8)))/((3)^(2) - (sqrt(8))^(2))- ((sqrt(8) + sqrt(7)))/((sqrt(8))^(2)-(sqrt(7))^(2)) +((sqrt(7) + sqrt(6)))/((sqrt(7))^(2) - (sqrt(6))^(2))-((sqrt(6) + sqrt(5)))/((sqrt(6))^(2) - (sqrt(5))^(2))+((sqrt(5) + 2))/((sqrt(5))^(2) - (2)^(2))`
`= (( 3+ sqrt(8)))/( (9-8))- ((sqrt(8)+ sqrt(7)))/((8-7)) + (sqrt(7)+ sqrt(6))/((7-6)) - (sqrt(6) + sqrt(5))/((6-5)) + (sqrt(5) + 2)/((5-4))`
` = (3+sqrt(8)) - (sqrt(8) + sqrt(7)) + (sqrt(7) + sqrt(6)) - (sqrt(6) + sqrt(5)) + (sqrt(5) + 2)`
` = 3+ sqrt(8) - sqrt(8) - sqrt(7) + sqrt(7) + sqrt(6) - sqrt(6) - sqrt(5) +sqrt(5) + 2`
` = 3+ 2 = 5`