Đáp án:
`a)\sqrt{\sqrt5-\sqrt{3-\sqrt{29-12\sqrt5}}}`
`=\sqrt{\sqrt5-\sqrt{3-\sqrt{20-2.2\sqrt5.3+9}}}`
`=\sqrt{\sqrt5-\sqrt{3-\sqrt{(2\sqrt5-3)^2}}}`
`=\sqrt{\sqrt5-\sqrt{3-(2\sqrt5-3)}}`
`=\sqrt{\sqrt5-\sqrt{3+3-2\sqrt5}}`
`=\sqrt{\sqrt5-\sqrt{6-2\sqrt5}}`
`=\sqrt{\sqrt5-\sqrt{5-2\sqrt5+1}}`
`=\sqrt{\sqrt5-\sqrt{(\sqrt5-1)^2}}`
`=\sqrt{\sqrt5-\sqrt5+1}`
`=\sqrt1=1`
`b)\sqrt{13+30\sqrt{2+\sqrt{9+4\sqrt2}}}`
`=\sqrt{13+30\sqrt{2+\sqrt{8+2.2\sqrt2+1}}}`
`=\sqrt{13+30\sqrt{2+\sqrt{(2\sqrt2+1)^2}}}`
`=\sqrt{13+30\sqrt{2+2\sqrt2+1}}`
`=\sqrt{13+30\sqrt{(\sqrt2+1)^2}}`
`=\sqrt{13+30(\sqrt2+1)}`
`=\sqrt{13+30\sqrt2+30}`
`=\sqrt{43+30\sqrt2}`
`=\sqrt{25+2.5.3\sqrt2+18}`
`=\sqrt{(5+3\sqrt2)^2}`
`=5+3\sqrt2`
`c)(\sqrt3-\sqrt2)\sqrt{5+2\sqrt6}`
`=(\sqrt3-\sqrt2)\sqrt{3+2\sqrt{2.3}+2}`
`=(\sqrt3-\sqrt2)\sqrt{(\sqrt3+\sqrt2)^2}`
`=(\sqrt3-\sqrt2)(\sqrt3+\sqrt2)`
`=3-2=1`
`d)\sqrt{5-\sqrt{13+4\sqrt3}}+\sqrt{3+\sqrt{13+4\sqrt3}}`
`=\sqrt{5-\sqrt{12+2.2\sqrt3+1}}+\sqrt{3+\sqrt{12+2.2\sqrt3+1}}`
`=\sqrt{5-\sqrt{(2\sqrt3+1)^2}}+\sqrt{3+\sqrt{(2\sqrt3+1)^2}}`
`=\sqrt{5-2\sqrt3-1}+\sqrt{3+2\sqrt3+1}`
`=\sqrt{3-2\sqrt3+1}+\sqrt{3+2\sqrt3+1}`
`=\sqrt{(\sqrt3-1)^2}+\sqrt{(\sqrt3+1)^2}`
`=\sqrt3-1+\sqrt3+1=2\sqrt3`
`e)\sqrt{1+\sqrt{3+\sqrt{13+4\sqrt3}}}+\sqrt{1-\sqrt{3-\sqrt{13-4\sqrt3}}}`
`=\sqrt{1+\sqrt{3+\sqrt{12+2.2\sqrt3+1}}}+\sqrt{1-\sqrt{3-\sqrt{12-2.2\sqrt3+1}}}`
`=\sqrt{1+\sqrt{3+\sqrt{(2\sqrt3+1)^2}}}+\sqrt{1-\sqrt{3-\sqrt{(2\sqrt3-1)^2}}}`
`=\sqrt{1+\sqrt{3+2\sqrt3+1}}+\sqrt{1-\sqrt{3-2\sqrt3+1}}`
`=\sqrt{1+\sqrt{(\sqrt3+1)^2}}+\sqrt{1-\sqrt{(\sqrt3-1)^2}}`
`=\sqrt{1+1+\sqrt3}+\sqrt{1-\sqrt3+1}`
`=\sqrt{2+\sqrt3}+\sqrt{2-\sqrt3}`
`=\sqrt{(4+2\sqrt3)/2}+\sqrt{(4-2\sqrt3)/2}`
`=\sqrt{(3+2\sqrt3+1)/2}+\sqrt{(3-2\sqrt3+1)/2}`
`=\sqrt{(\sqrt3+1)^2/2}+\sqrt{(\sqrt3-1)^2/2}`
`=(\sqrt3+1)/\sqrt2+(\sqrt3-1)/\sqrt2`
`=(\sqrt3+1+\sqrt3-1)/\sqrt2`
`=(2\sqrt3)/\sqrt2=\sqrt6`
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