`1)` `\sqrt{11-6\sqrt{2}}-\sqrt{11+6\sqrt{2}}`
`=\sqrt{11-2.3\sqrt{2}}-\sqrt{11+2.3\sqrt{2}}`
`=\sqrt{11-2\sqrt{2.9}}-\sqrt{11+2\sqrt{2.9}}`
`=\sqrt{2-2\sqrt{2.9}+9}-\sqrt{2+2\sqrt{2.9}+9}`
`=\sqrt{(\sqrt{2}-3)^2}-\sqrt{(\sqrt{2}+3)^2}`
`=|\sqrt{2}-3|-|\sqrt{2}+3|`
`=3-\sqrt{2}-(\sqrt{2}+3)` (do `\sqrt{2}-3<0;\sqrt{2}+3>0)`
`=3-\sqrt{2}-\sqrt{2}-3`
`=-2\sqrt{2}`
`2)` `\sqrt{(2-\sqrt{5})^2}+\sqrt{14-6\sqrt{5}}`
`=|2-\sqrt{5}|+\sqrt{14-2.3\sqrt{5}}`
`=\sqrt{5}-2+\sqrt{14-2\sqrt{5.9}}` (do `2-\sqrt{5}<0)`
`=\sqrt{5}-2+\sqrt{5-2\sqrt{5.9}+9}`
`=\sqrt{5}-2+\sqrt{(\sqrt{5}-3)^2}`
`=\sqrt{5}-2+|\sqrt{5}-3|`
`=\sqrt{5}-2+(3-\sqrt{5})` (do `\sqrt{5}-3<0)`
`=\sqrt{5}-2+3-\sqrt{5}`
`=1`
`3)` `(2+\sqrt{7})\sqrt{11-4\sqrt{7}}`
`=(2+\sqrt{7})\sqrt{11-2.2\sqrt{7}}`
`=(2+\sqrt{7})\sqrt{11-2\sqrt{7.4}}`
`=(2+\sqrt{7})\sqrt{7-2\sqrt{7.4}+4}`
`=(2+\sqrt{7})\sqrt{(\sqrt{7}-2)^2}`
`=(2+\sqrt{7}|\sqrt{7}-2|`
`=(2+\sqrt{7})(\sqrt{7}-2)` (do `\sqrt{7}-2>0)`
`=7-4`
`=3`
`4)` `\sqrt{(3+\sqrt{2})^2}+\sqrt{6-4\sqrt{2}}`
`=|3+\sqrt{2}|+\sqrt{6-2.2\sqrt{2}}`
`=3+\sqrt{2}+\sqrt{6-2\sqrt{2.4}}` (do `3+\sqrt{2}>0)`
`=3+\sqrt{2}+\sqrt{2-2\sqrt{2.4}+4}`
`=3+\sqrt{2}+\sqrt{(\sqrt{2}-2)^2}`
`=3+\sqrt{2}+|\sqrt{2}-2|`
`=3+\sqrt{2}+(2-\sqrt{2})` (do `\sqrt{2}-2<0)`
`=3+\sqrt{2}+2-\sqrt{2}`
`=5`
