`1/(2+\sqrt{5}+2\sqrt{2}+\sqrt{10})`

`=(2+\sqrt{5}-2\sqrt{2}-\sqrt{10})/[(2+\sqrt{5}+2\sqrt{2}+\sqrt{10})(2+\sqrt{5}-2\sqrt{2}-\sqrt{10})]`

`=(2+\sqrt{5}-2\sqrt{2}-\sqrt{10})/[(2+\sqrt{5})^2-(2\sqrt{2}+\sqrt{10})^2]`

`=(2+\sqrt{5}-2\sqrt{2}-\sqrt{10})/[4+4\sqrt{5}+5-(8+8\sqrt{5}+10)]`

`=(2+\sqrt{5}-2\sqrt{2}-\sqrt{10})/[4+4\sqrt{5}+5-8-8\sqrt{5}-10]`

`=(2+\sqrt{5}-2\sqrt{2}-\sqrt{10})/[-9-4\sqrt{5}]`

`=[(2+\sqrt{5}-2\sqrt{2}-\sqrt{10})(-9+4\sqrt{5})]/[(-9-4\sqrt{5})(-9+4\sqrt{5})]`

`=[(2+\sqrt{5}-2\sqrt{2}-\sqrt{10})(-9+4\sqrt{5})]/[81-16.5]`

`=(2+\sqrt{5}-2\sqrt{2}-\sqrt{10})(-9+4\sqrt{5})`

`=-18+8\sqrt{5}-9sqrt5+20+18sqrt2-8\sqrt{10}+9\sqrt{10}-4\sqrt{50}`

`=2-\sqrt{5}-2\sqrt{2}+\sqrt{10}`