`5)``\sqrt{23 + 4\sqrt{15}} = \sqrt{23 + 2\sqrt{60}}`

`= \sqrt{(\sqrt{20})^2 + 2\cdot\sqrt{20}\cdot\sqrt{3} + (\sqrt{3})^2}`

`= \sqrt{(\sqrt{20} + \sqrt{3})^2}`

`= |\sqrt{20} + \sqrt{3}|`

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

`6)``\sqrt{16 - 2\sqrt{55}} = \sqrt{(\sqrt{11})^2 - 2\cdot\sqrt{11}\cdot\sqrt{5} + (\sqrt{5})^2}`

`= \sqrt{(\sqrt{11} - \sqrt{5})^2}`

`= |\sqrt{11} - \sqrt{5}|`

`= \sqrt{11} - \sqrt{5}`

`7)``\sqrt{8 - 2\sqrt{12}} = \sqrt{(\sqrt{6})^2 - 2\cdot\sqrt{6}\cdot\sqrt{2} + (\sqrt{2})^2}`

`= \sqrt{(\sqrt{6} - \sqrt{2})^2}`

`= |\sqrt{6} - \sqrt{2}|`

`= \sqrt{6} - \sqrt{2}`

`8)``\sqrt{8 + 2\sqrt{15}} = \sqrt{(\sqrt{5})^2 + 2\cdot\sqrt{5}\cdot\sqrt{3} + (\sqrt{3})^2}`

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

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

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