\(\sqrt{21+6\sqrt{6}}\)
\(=\sqrt{21+2\cdot\sqrt{108}}\)
\(=2\sqrt{3}+3\)
\(\sqrt{29-12\sqrt{5}}\)
\(=\sqrt{29-2\cdot\sqrt{180}}\)
\(=2\sqrt{5}-3\)
\(\sqrt{36-12\sqrt{5}}\)
\(=\sqrt{36-2\cdot\sqrt{180}}\)
\(=\sqrt{30}-\sqrt{6}\)
1) \(\sqrt{21+6\sqrt{6}}=\sqrt{\left(3\sqrt{2}\right)^2+2.3\sqrt{2}.\sqrt{3}+3}=\sqrt{\left(3\sqrt{2}+\sqrt{3}\right)^2}=3\sqrt{2}+\sqrt{3}\)
2) \(\sqrt{29-12\sqrt{5}}=\sqrt{\left(20-12\sqrt{5}+9\right)}=\sqrt{\left(2\sqrt{5}-3\right)^2}=\left|2\sqrt{5}-3\right|=2\sqrt{5}-3\)
3) \(\sqrt{36-12\sqrt{5}}=\sqrt{\left(\sqrt{30}-\sqrt{6}\right)^2}=\left|\sqrt{30}-\sqrt{6}\right|=\sqrt{30}-\sqrt{6}\)