Đáp án+Giải thích các bước giải:

 `i)\sqrt{(2+\sqrt{5})^{2}}+\sqrt[21-8\sqrt{5}}`

`=|2+\sqrt{5}|+\sqrt{4^{2}-2.4.\sqrt{5}+(\sqrt{5})^{2}}`

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

`=2+\sqrt{5}+|4-\sqrt{5}|`

`=2+\sqrt{5}+4-\sqrt{5}`

`=6`

`j)\sqrt{(3+\sqrt{2})^{2}}+\sqrt{6-4\sqrt{2}}`

`=|3+\sqrt{2}|+\sqrt{2^{2}-2.2.\sqrt{2}+(\sqrt{2})^{2}}`

`=3+\sqrt{2}+\sqrt{(2-\sqrt{2})^{2}}`

`=3+\sqrt{2}+|2-\sqrt{2}|`

`=3+\sqrt{2}+2-\sqrt{2}`

`=5`

`k)A=2+\sqrt{17-4\sqrt{9+4\sqrt{5}}}`

`=2+\sqrt[17-4\sqrt{(\sqrt{5})^{2}+2.\sqrt{5}.2+2^{2}}}`

`=2+\sqrt[17-4\sqrt{(\sqrt{5}+2)^{2}}}`

`=2+\sqrt[17-4.|\sqrt{5}+2|}`

`=2+\sqrt[17-4.(\sqrt{5}+2)}`

`=2+\sqrt{17-4\sqrt{5}-8}`

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

`=2+\sqrt{(\sqrt{5})^{2}-2.\sqrt{5}.2+2^{2}}`

`=2+\sqrt{(\sqrt{5}-2)^{2}}`

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

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

`=\sqrt{5}`

vậy `A=\sqrt{5}`

`l)(\sqrt{15}-\sqrt{5})/(\sqrt{3}-1)+\sqrt{12-2\sqrt{35}}`

`=(\sqrt{5}.(\sqrt{3}-1))/(\sqrt{3}-1)+\sqrt{(\sqrt{7})^{2}-2.\sqrt{7}.\sqrt{5}+(\sqrt{5})^{2}}`

`=\sqrt{5}+\sqrt{(\sqrt{7}-\sqrt{5})^{2}}`

`=\sqrt{5}+|\sqrt{7}-\sqrt{5}|`

`=\sqrt{5}+\sqrt{7}-\sqrt{5}`

`=\sqrt{7}`

`m)\sqrt{(3-\sqrt{7})^{2}}+\sqrt{11+4\sqrt{7}}`

`=|3-\sqrt{7}|+\sqrt{2^{2}+2.2.\sqrt{7}+(\sqrt{7})^{2}}`

`=3-\sqrt{7}+\sqrt{(2+\sqrt{7})^{2}}`

`=3-\sqrt{7}+|2+\sqrt{7}|`

`=3-\sqrt{7}+2+\sqrt{7}`

`=5`

`n)\sqrt{28-16\sqrt{3}}+\sqrt{13-4\sqrt{3}}`

`=\sqrt{(2\sqrt{3})^{2}-2.2\sqrt{3}.4+4^{2}}+\sqrt{(2\sqrt{3})^{2}-2.2\sqrt{3}.1+1^{2}}`

`=\sqrt{(2\sqrt{3}-4)^{2}}+\sqrt{(2\sqrt{3}-1)^{2}}`

`=|2\sqrt{3}-4|+|2\sqrt{3}-1|`

`=4-2\sqrt{3}+2\sqrt{3}-1` (do `2\sqrt{3}-4<0;2\sqrt{3}-1>0`)

`=3`

`o)\sqrt{11+2\sqrt[10}}-\sqrt{12-2\sqrt[20}}`

`=\sqrt{(\sqrt{10})^{2}+2.\sqrt[10}.1+1^{2}}-\sqrt{(\sqrt[10})^{2}-2.\sqrt[10}.\sqrt{2}+(\sqrt{2})^{2}}`

`=\sqrt{(\sqrt[10}+1)^{2}}-\sqrt{(\sqrt[10}-\sqrt{2})^{2}}`

`=|\sqrt[10}+1|-|\sqrt[10}-\sqrt{2}|`

`=\sqrt[10}+1-(\sqrt[10}-\sqrt{2})`

`=\sqrt[10}+1-\sqrt{10}+\sqrt{2}`

`=1+\sqrt{2}`