Đáp án:

$65. \sqrt{2}+\sqrt{3}+\sqrt{5}\\ 66. \sqrt{2}-\sqrt{3}+\sqrt{5}\\ 67. \sqrt{2}+\sqrt{3}-\sqrt{5}\\ 68. \sqrt{3}-\sqrt{2}+\sqrt{5}$

Giải thích các bước giải:

$65. \sqrt{10+2\sqrt{6}+2\sqrt{10}+2\sqrt{15}}\\ =\sqrt{2+2\sqrt{6}+3+5+2\sqrt{10}+2\sqrt{15}}\\ =\sqrt{2+2\sqrt{2}.\sqrt{3}+3+5+2\sqrt{10}+2\sqrt{15}}\\ =\sqrt{(\sqrt{2}+\sqrt{3})^2+2\sqrt{10}+2\sqrt{15}+5}\\ =\sqrt{(\sqrt{2}+\sqrt{3})^2+2\sqrt{5}(\sqrt{2}+2\sqrt{3})+5}\\ =\sqrt{(\sqrt{2}+\sqrt{3}+\sqrt{5})^2}\\ =\sqrt{2}+\sqrt{3}+\sqrt{5}\\ 66. \sqrt{10-2\sqrt{6}+2\sqrt{10}-2\sqrt{15}}\\ =\sqrt{2-2\sqrt{6}+3+5+2\sqrt{10}-2\sqrt{15}}\\ =\sqrt{2-2\sqrt{2}.\sqrt{3}+3+5+2\sqrt{10}-2\sqrt{15}}\\ =\sqrt{(\sqrt{2}-\sqrt{3})^2+2\sqrt{10}-2\sqrt{15}+5}\\ =\sqrt{(\sqrt{2}-\sqrt{3})^2+2\sqrt{5}(\sqrt{2}-\sqrt{3})+5}\\ =\sqrt{(\sqrt{2}-\sqrt{3}+\sqrt{5})^2}\\ =\sqrt{2}-\sqrt{3}+\sqrt{5}\\ 67. \sqrt{10+2\sqrt{6}-2\sqrt{10}-2\sqrt{15}}\\ =\sqrt{2+2\sqrt{6}+3+5-2\sqrt{10}-2\sqrt{15}}\\ =\sqrt{2+2\sqrt{2}.\sqrt{3}+3+5-2\sqrt{10}-2\sqrt{15}}\\ =\sqrt{(\sqrt{2}+\sqrt{3})^2-2\sqrt{10}-2\sqrt{15}+5}\\ =\sqrt{(\sqrt{2}+\sqrt{3})^2-2\sqrt{5}(\sqrt{2}+\sqrt{3})+5}\\ =\sqrt{(\sqrt{2}+\sqrt{3}-\sqrt{5})^2}\\ =\sqrt{2}+\sqrt{3}-\sqrt{5}\\ 68. \sqrt{10-2\sqrt{6}-2\sqrt{10}+2\sqrt{15}}\\ =\sqrt{2-2\sqrt{6}+3+5-2\sqrt{10}+2\sqrt{15}}\\ =\sqrt{2-2\sqrt{2}.\sqrt{3}+3+5-2\sqrt{10}+2\sqrt{15}}\\ =\sqrt{(\sqrt{2}-\sqrt{3})^2-2\sqrt{10}+2\sqrt{15}+5}\\ =\sqrt{(\sqrt{2}-\sqrt{3})^2+2\sqrt{5}(\sqrt{3}-\sqrt{2})+5}\\ =\sqrt{(\sqrt{3}-\sqrt{2})^2+2\sqrt{5}(\sqrt{3}-\sqrt{2})+5}\\ =\sqrt{(\sqrt{3}-\sqrt{2}+\sqrt{5})^2}\\ =\sqrt{3}-\sqrt{2}+\sqrt{5}$

Đáp án + Giải thích các bước giải:

`65. \sqrt{10+2\sqrt{6}+2\sqrt{10}+2\sqrt{15}}`

`=\sqrt{2+2\sqrt{6}+3+5+2\sqrt{10}+2\sqrt{15}}`

`=\sqrt{2+2\sqrt{2}.\sqrt{3}+3+5+2\sqrt{10}+2\sqrt{15}}`

`=\sqrt{(\sqrt{2}+\sqrt{3})^2+2\sqrt{10}+2\sqrt{15}+5}`

`=\sqrt{(\sqrt{2}+\sqrt{3})^2+2\sqrt{5}(\sqrt{2}+2\sqrt{3})+5}`

`=\sqrt{(\sqrt{2}+\sqrt{3}+\sqrt{5})^2}`

`=\sqrt{2}+\sqrt{3}+\sqrt{5}`

`66. \sqrt{10-2\sqrt{6}+2\sqrt{10}-2\sqrt{15}}`

`=\sqrt{2-2\sqrt{6}+3+5+2\sqrt{10}-2\sqrt{15}}`

`=\sqrt{2-2\sqrt{2}.\sqrt{3}+3+5+2\sqrt{10}-2\sqrt{15}}`

`=\sqrt{(\sqrt{2}-\sqrt{3})^2+2\sqrt{10}-2\sqrt{15}+5}`

`=\sqrt{(\sqrt{2}-\sqrt{3})^2+2\sqrt{5}(\sqrt{2}-\sqrt{3})+5}`

`=\sqrt{(\sqrt{2}-\sqrt{3}+\sqrt{5})^2}`

`=\sqrt{2}-\sqrt{3}+\sqrt{5}`

`67. \sqrt{10+2\sqrt{6}-2\sqrt{10}-2\sqrt{15}}`

`=\sqrt{2+2\sqrt{6}+3+5-2\sqrt{10}-2\sqrt{15}}`

`=\sqrt{2+2\sqrt{2}.\sqrt{3}+3+5-2\sqrt{10}-2\sqrt{15}}`

`=\sqrt{(\sqrt{2}+\sqrt{3})^2-2\sqrt{10}-2\sqrt{15}+5}`

`=\sqrt{(\sqrt{2}+\sqrt{3})^2-2\sqrt{5}(\sqrt{2}+\sqrt{3})+5}`

`=\sqrt{(\sqrt{2}+\sqrt{3}-\sqrt{5})^2}`

`=\sqrt{2}+\sqrt{3}-\sqrt{5}`

`68. \sqrt{10-2\sqrt{6}-2\sqrt{10}+2\sqrt{15}}`

`=\sqrt{2-2\sqrt{6}+3+5-2\sqrt{10}+2\sqrt{15}}`

`=\sqrt{2-2\sqrt{2}.\sqrt{3}+3+5-2\sqrt{10}+2\sqrt{15}}`

`=\sqrt{(\sqrt{2}-\sqrt{3})^2-2\sqrt{10}+2\sqrt{15}+5}`

`=\sqrt{(\sqrt{2}-\sqrt{3})^2+2\sqrt{5}(\sqrt{3}-\sqrt{2})+5}`

`=\sqrt{(\sqrt{3}-\sqrt{2})^2+2\sqrt{5}(\sqrt{3}-\sqrt{2})+5}`

`=\sqrt{(\sqrt{3}-\sqrt{2}+\sqrt{5})^2}`

`=\sqrt{3}-\sqrt{2}+\sqrt{5} `