Đáp án:
`a)` `\sqrt{9-\sqrt{17}}-\sqrt{9+\sqrt{17}}=-\sqrt{2}`
`b)` `\sqrt{11-2\sqrt{30}}+\sqrt{13+2\sqrt{42}}=\sqrt{7}+2\sqrt{6}-\sqrt{5}`
Giải thích các bước giải:
`a)` `\sqrt{9-\sqrt{17}}-\sqrt{9+\sqrt{17}}`
`=\sqrt{{18-2\sqrt{17}}/2}-\sqrt{{18+2\sqrt{17}}/2}`
`=\sqrt{{17-2.\sqrt{17}.1+1^2}/2}-\sqrt{{17+2.\sqrt{17}.1+1^2}/2}`
`=\sqrt{{(\sqrt{17}-1)^2}/2}-\sqrt{{(\sqrt{17}+1)^2}/2}`
`=|{\sqrt{17}-1}/\sqrt{2}|-|{\sqrt{17}+1}/\sqrt{2}|`
`={\sqrt{17}-1}/\sqrt{2}-{\sqrt{17}+1}/\sqrt{2}`
`={\sqrt{17}-1-\sqrt{17}-1}/\sqrt{2}={-2}/\sqrt{2}=-\sqrt{2}`
Vậy: `\sqrt{9-\sqrt{17}}-\sqrt{9+\sqrt{17}}=-\sqrt{2}`
$\\$
`b)` `\sqrt{11-2\sqrt{30}}+\sqrt{13+2\sqrt{42}}`
`=\sqrt{6-2.\sqrt{6}.\sqrt{5}+5}+\sqrt{7+2.\sqrt{7}.\sqrt{6}+6}`
`=\sqrt{(\sqrt{6}-\sqrt{5})^2}+\sqrt{(\sqrt{7}+\sqrt{6})^2}`
`=|\sqrt{6}-\sqrt{5}|+|\sqrt{7}+\sqrt{6}|`
`=\sqrt{6}-\sqrt{5}+\sqrt{7}+\sqrt{6}`
`=\sqrt{7}+2\sqrt{6}-\sqrt{5}`
Vậy: `\sqrt{11-2\sqrt{30}}+\sqrt{13+2\sqrt{42}}=\sqrt{7}+2\sqrt{6}-\sqrt{5}`
