Question

Answer 1

`a)\sqrt{3+2\sqrt{2}}-\sqrt{6-4\sqrt{2}}`

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

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

`b)\sqrt{\sqrt{5}-\sqrt{3-\sqrt{29-12\sqrt{5}}}}`

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

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

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

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

`c)\sqrt{6+2\sqrt{5}-\sqrt{29-12\sqrt{5}}}`

`=\sqrt{6+2\sqrt{5}-2\sqrt{5}+3}`

`=\sqrt{9}=3`

`d)\sqrt{2+\sqrt{5-\sqrt{13+\sqrt{48}}}}`

`=\sqrt{2+\sqrt{5-\sqrt{13+4\sqrt{3}}}}`

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

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

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

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

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