`[2\sqrt{3+\sqrt{5-\sqrt{13+\sqrt{48}}}}]/[\sqrt{6}-\sqrt{2}]`

Đáp án :

Xét tử:

`2\sqrt{3 + \sqrt{5 - \sqrt{ 13 + \sqrt{48}} } }`

 ` = 2\sqrt{3 + \sqrt{5 - \sqrt{12 + 2\sqrt{12} + 1} } }`

` = 2\sqrt{3 + \sqrt{5 - \sqrt{ (\sqrt{12} + 1)^2} } }`

` = 2\sqrt{3 +  \sqrt{5 - |2\sqrt{3} + 1|} }`

` = 2\sqrt{3 + \sqrt{5 - 2\sqrt{3} - 1} }`

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

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

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

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

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

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

` = \sqrt{2} . \sqrt{4 + 2\sqrt{3}}`

` = \sqrt{2} . \sqrt{3 + 2\sqrt{3} + 1}`

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

` = \sqrt{2} . |\sqrt{3} + 1|`

` = \sqrt{2} . (\sqrt{3} + 1)`

`=> (\sqrt{2} . (\sqrt{3} + 1))/(\sqrt{2}(\sqrt{3} - 1))`

` = (\sqrt{3} + 1)/(\sqrt{3} - 1)`

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

` = (4 + 2\sqrt{3})/2`

` = 2 + \sqrt{3}`