`1/(2+sqrt5+2sqrt2+sqrt10)`
`=1/((2+sqrt5)+2sqrt(2+sqrt5))`
`=1/((2+sqrt5)(1+sqrt2))`
`=((2-sqrt5)(1-sqrt2))/((2+sqrt5)(2-sqrt5)(1-sqrt2)(1+sqrt2))`
`=((2-sqrt5)(1-sqrt2))/([2^2-(sqrt5)^2][1^2-(sqrt2)^2])`
`=(2-2sqrt2-sqrt5+sqrt10)/((4-5)(1-2))`
`=(2-2sqrt2-sqrt5+sqrt10)/((-1)*(-1))`
`=(2-2sqrt2-sqrt5+sqrt10)/1`
`=2-2sqrt2-sqrt5+sqrt10`