Đáp án:
`a)` `\sqrt{5}+1`
`b)` `1`
`c)` `\sqrt{5}-1`
`d)` `5+3\sqrt{2}`
Giải thích các bước giải:
`a)` `\sqrt{8+2\sqrt{6-\sqrt{20}}}`
`=\sqrt{8+2\sqrt{6-2\sqrt{5}}}`
`=\sqrt{8+2\sqrt{5-2\sqrt{5}.1+1^2}}`
`=\sqrt{8+2\sqrt{(\sqrt{5}-1)^2}}`
`=\sqrt{8+2.|\sqrt{5}-1|}`
`=\sqrt{8+2(\sqrt{5}-1)}` `=\sqrt{6+2\sqrt{5}}`
`=\sqrt{5+2\sqrt{5}.1+1^2}`
`=\sqrt{(\sqrt{5}+1)^2}`
`=|\sqrt{5}+1|=\sqrt{5}+1`
Vậy: `\sqrt{8+2\sqrt{6-\sqrt{20}}}=\sqrt{5}+1`
$\\$
`b)` `\sqrt{\sqrt{5}-\sqrt{6-2\sqrt{5}}}`
`=\sqrt{\sqrt{5}-\sqrt{5-2\sqrt{5}.1+1^2}}`
`=\sqrt{\sqrt{5}-\sqrt{(\sqrt{5}-1)^2}}`
`=\sqrt{\sqrt{5}-|\sqrt{5}-1|}`
`=\sqrt{\sqrt{5}-(\sqrt{5}-1)}`
`=\sqrt{1}=1`
Vậy: `\sqrt{\sqrt{5}-\sqrt{6-2\sqrt{5}}}=1`
$\\$
`c)` `\sqrt{3-\sqrt{29-12\sqrt{5}}}=\sqrt{3-\sqrt{20-12\sqrt{5}+9}}`
`=\sqrt{3-\sqrt{(2\sqrt{5})^2-2.2\sqrt{5}.3+3^2}}`
`=\sqrt{3-\sqrt{(2\sqrt{5}-3)^2}}`
`=\sqrt{3-|2\sqrt{5}-3|}=\sqrt{3-(2\sqrt{5}-3)}`
`=\sqrt{6-2\sqrt{5}}=\sqrt{5-2\sqrt{5}.1+1^2}`
`=\sqrt{(\sqrt{5}-1)^2}=|\sqrt{5}-1|`
`=\sqrt{5}-1`
Vậy: `\sqrt{3-\sqrt{29-12\sqrt{5}}}=\sqrt{5}-1`
$\\$
`d)` `\sqrt{13+30\sqrt{3+2\sqrt{2}}}`
`=\sqrt{13+30\sqrt{2+2\sqrt{2}.1+1^2}}`
`=\sqrt{13+30\sqrt{(\sqrt{2}+1)^2}}`
`=\sqrt{13+30.|\sqrt{2}+1|}`
`=\sqrt{13+30.(\sqrt{2}+1)}=\sqrt{43+30\sqrt{2}}`
`=\sqrt{25+30\sqrt{2}+18}`
`=\sqrt{5^2+2.5.3\sqrt{2}+(3\sqrt{2})^2}`
`=\sqrt{(5+3\sqrt{2})^2}=|5+3\sqrt{2}|`
`=5+3\sqrt{2}`
Vậy: `\sqrt{13+30\sqrt{3+2\sqrt{2}}}=5+3\sqrt{2}`
