ta có \(\left(\sqrt{5\sqrt{3}}\right)^4=75\)

          \(\left(\sqrt{3\sqrt{5}}\right)^4=45\)

 \(\Rightarrow\sqrt{5\sqrt{3}}>\sqrt{3\sqrt{5}}\left(75>45\right)\)

Mình thấy cả 2 con trên đều bằng nhau .

\(\sqrt{5\sqrt{3}}\)và \(\sqrt{3\sqrt{5}}\)

\(\sqrt{5\sqrt{3}}\)= 2,94283................

 \(\sqrt{3\sqrt{5}}\)= 2,59002...............

Ta thấy 2,94283......... > 2,59002.........Suy ra \(\sqrt{5\sqrt{3}}\) > \(\sqrt{3\sqrt{5}}\)

\(\sqrt{\sqrt{6+\sqrt{20}}}=\sqrt{\sqrt{5+2\sqrt{5}+1}}=\sqrt{\sqrt{\left(\sqrt{5}+1\right)^2}}=\sqrt{\sqrt{5}+1}< \sqrt{\sqrt{6}+1}\)

\(\sqrt{2}\)+3=3+\(\sqrt{2}\)

\(\sqrt{3}\)+2=2+\(\sqrt{3}\)

\(\Rightarrow\)\(\sqrt{2}\)+3>\(\sqrt{3}\)+2

m kmnhbk5htb ,k55555555555555555555555555555555555e,

\(\sqrt{\sqrt{6+\sqrt{20}}}=\sqrt{\sqrt{6+2\sqrt{5}}}=\sqrt{\sqrt{\left(\sqrt{5}+1\right)^2}}=\sqrt{\sqrt{5}+1}\)

Vì \(\sqrt{\sqrt{5}+1}< \sqrt{\sqrt{6}+1}\Rightarrow\sqrt{\sqrt{6+\sqrt{20}}}< \sqrt{1+\sqrt{6}}\)

Có \(\sqrt{\sqrt{6+\sqrt{20}}}=\sqrt{\sqrt{5+2\sqrt{5}+1}}\)

                                           \(=\sqrt{\sqrt{\left(1+\sqrt{5}\right)^2}}\)

                                          \(=\sqrt{1+\sqrt{5}}< \sqrt{1+\sqrt{6}}\)

Vậy \(\sqrt{\sqrt{6+\sqrt{20}}}< \sqrt{1+\sqrt{6}}\)