`s=2\sqrt(3+\sqrt(5-\sqrt(13)+\sqrt(48)))`

$\displaystyle \begin{array}{{>{\displaystyle}l}} 2\sqrt{3+\sqrt{5-\sqrt{13+\sqrt{48}}}}\\ =2\sqrt{3+\sqrt{5-\sqrt{13+4\sqrt{3}}}}\\ =2\sqrt{3+\sqrt{5-\sqrt{12+2.2\sqrt{3} +1}}}\\ =2\sqrt{3+\sqrt{5-\sqrt{\left( 2\sqrt{3} +1\right)^{2}}}}\\ =2\sqrt{3+\sqrt{5-2\sqrt{3} -1}}\\ =2\sqrt{3+\sqrt{4-2\sqrt{3}}}\\ =2\sqrt{3+\sqrt{3-2\sqrt{3} +1}}\\ =2\sqrt{3+\sqrt{\left(\sqrt{3} -1\right)^{2}}}\\ =2\sqrt{3+\sqrt{3} -1}\\ =2\sqrt{2+\sqrt{3}} \ \\ =\sqrt{8+4\sqrt{3}} \ \\ =\sqrt{8+2.\sqrt{12}}\\ =\sqrt{2+2.\sqrt{2} .\sqrt{6} +6}\\ =\sqrt{\left(\sqrt{2} +\sqrt{6}\right)^{2}} =\sqrt{2} +\sqrt{6} \ \end{array}$

Đáp án và giải thích các bước giải:

Có : `\sqrt[48]=4\sqrt[3]`

`\sqrt[13+\sqrt[48]]=\sqrt[13+4\sqrt[3]]=\sqrt[(2\sqrt[3]+1)^2]=|2\sqrt[3]+1|=2\sqrt[3]+1`

`\sqrt[5-\sqrt[13]+\sqrt[48]]=\sqrt[5-2\sqrt[3]-1]=\sqrt[4-2\sqrt[3]]=\sqrt[(\sqrt[3]-1)^2]=|\sqrt[3]-1|=\sqrt[3]-1`

`⇒` `s=2\sqrt[3+\sqrt[5-\sqrt[13]+\sqrt[48]]]`

`=2\sqrt[3+(\sqrt[3]-1)]=\sqrt[2(4+2\sqrt[3])]=(\sqrt[3]+1)\sqrt[2]=\sqrt[6]+\sqrt[2]`