Question

\(\sqrt{17-4\sqrt{9+4\sqrt{5}}}\)

Answer 1

Ta có : \(\sqrt{17-4\sqrt{9+4\sqrt{5}}}\)\(=\sqrt{17-4\sqrt{4+2.2\sqrt{5}+5}}\)

\(=\sqrt{17-4\sqrt{\left(2+\sqrt{5}\right)^2}}=\sqrt{17-4\left(2+\sqrt{5}\right)}=\sqrt{17-8-4\sqrt{5}}\)

\(=\sqrt{9-4\sqrt{5}}=\sqrt{4-2.2\sqrt{5}+5}=\sqrt{\left(2-\sqrt{5}\right)^2}=\sqrt{5}-2\)

 

Answer 2

`sqrt{17-4sqrt{9+4sqrt5}}`

`=sqrt{17-4sqrt{5+2.2.sqrt5+4}}`

`=sqrt{17-4sqrt{(2+sqrt5)^2}}`

`=sqrt{17-4(2+sqrt5)}`

`=sqrt{17-8-4sqrt5}`

`=sqrt{9-4sqrt5}`

`=sqrt{5-2.2.sqrt5+4}`

`=sqrt{(sqrt5-2)^2}`

`=|sqrt5-2|`

`=sqrt5-2`