1:
\(\sqrt{\sqrt{5}-\sqrt{3-\sqrt{29-12\sqrt{5}}}}\)
\(=\sqrt{\sqrt{5}-\sqrt{3-\left(2\sqrt{5}-3\right)}}\)
\(=\sqrt{\sqrt{5}-\sqrt{3-2\sqrt{5}+3}}\)
\(=\sqrt{\sqrt{5}-\sqrt{6-2\sqrt{5}}}\)
\(=\sqrt{\sqrt{5}-\left(\sqrt{5}-1\right)}=\sqrt{1}=1\)
2: \(\sqrt{13+30\sqrt{2+\sqrt{9+4\sqrt{2}}}}\)
\(=\sqrt{13+30\sqrt{2+\left(2\sqrt{2}+1\right)}}\)
\(=\sqrt{13+30\sqrt{2+2\cdot\sqrt{2}\cdot1}}\)
\(=\sqrt{13+30\left(\sqrt{2}+1\right)}\)
\(=\sqrt{43+30\sqrt{2}}\)
\(=\sqrt{25+2\cdot5\cdot3\sqrt{2}+18}\)
\(=\sqrt{\left(5+3\sqrt{2}\right)^2}=5+3\sqrt{2}\)
3: \(\sqrt{6+2\sqrt{5-\sqrt{13+\sqrt{48}}}}\)
\(=\sqrt{6+2\sqrt{5-\sqrt{13+4\sqrt{3}}}}\)
\(=\sqrt{6+2\sqrt{5-\left(2\sqrt{3}+1\right)}}\)
\(=\sqrt{6+2\sqrt{5-2\sqrt{3}-1}}\)
\(=\sqrt{6+2\sqrt{4-2\sqrt{3}}}\)
\(=\sqrt{6+2\left(\sqrt{3}-1\right)}\)
\(=\sqrt{6+2\sqrt{3}-2}=\sqrt{4+2\sqrt{3}}=\sqrt{3}+1\)
4: \(\sqrt{4+\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{7+4\sqrt{3}}}}}\)
\(=\sqrt{4+\sqrt{5\sqrt{3}+5\sqrt{48-10\left(2+\sqrt{3}\right)}}}\)
\(=\sqrt{4+\sqrt{5\sqrt{3}+5\sqrt{48-20-10\sqrt{3}}}}\)
\(=\sqrt{4+\sqrt{5\sqrt{3}+5\sqrt{28-10\sqrt{3}}}}\)
\(=\sqrt{4+\sqrt{5\sqrt{3}+5\left(5-\sqrt{3}\right)}}\)
\(=\sqrt{4+\sqrt{5\sqrt{3}+25-5\sqrt{3}}}\)
\(=\sqrt{4+\sqrt{25}}=\sqrt{4+5}=3\)
5:
\(\sqrt{3-2\sqrt{2}}+\sqrt{5-2\sqrt{6}}+...+\sqrt{19-2\sqrt{90}}\)
\(=-1+\sqrt{2}-\sqrt{2}+\sqrt{3}-\sqrt{3}+...-\sqrt{9}+\sqrt{10}\)
\(=\sqrt{10}-1\)
6:
\(\sqrt{\left(\sqrt{5}-1\right)\sqrt{13-\sqrt{69-28\sqrt{5}}}}\)
\(=\sqrt{\left(\sqrt{5}-1\right)\cdot\sqrt{13-\sqrt{49-2\cdot7\cdot2\sqrt{5}+20}}}\)
\(=\sqrt{\left(\sqrt{5}-1\right)\cdot\sqrt{13-\left(7-2\sqrt{5}\right)}}\)
\(=\sqrt{\left(\sqrt{5}-1\right)\cdot\sqrt{6+2\sqrt{5}}}\)
\(=\sqrt{\left(\sqrt{5}-1\right)\left(\sqrt{5}+1\right)}=\sqrt{5-1}=2\)
