Question

So sánh :

a) \(\sqrt{\sqrt{6+\sqrt{20}}}\) và \(\sqrt{1+\sqrt{6}}\)

b) \(\sqrt{\sqrt{17+12\sqrt{2}}}\) và \(\sqrt{2}+1\)

c) \(\sqrt{\sqrt{28-16\sqrt{3}}}\) và \(\sqrt{3}-2\)

Answer 1

a ) VT = \(\sqrt{\sqrt{6+\sqrt{20}}}=\sqrt{\sqrt{6+\sqrt{4.5}}}=\sqrt{\sqrt{6+2\sqrt{5}}}=\sqrt{\sqrt{\left(1+\sqrt{5}\right)^2}}=\sqrt{1+\sqrt{5}}\)

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

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

b) VT = \(\sqrt{\sqrt{17+12\sqrt{2}}}=\sqrt{\sqrt{17+2.2\sqrt{2}.3}}=\sqrt{\sqrt{\left(2\sqrt{2}+3\right)^2}=\sqrt{2\sqrt{2}+3}}=\sqrt{\left(\sqrt{2}+1\right)^2}=\sqrt{2}+1\)

=> VT = VP

=> \(\sqrt{\sqrt{17+12\sqrt{2}}}=\sqrt{2}+1\)

c) \(\sqrt{\sqrt{28-16\sqrt{3}}}=\sqrt{\sqrt{16-2.4.2\sqrt{3}+12}}=\sqrt{\sqrt{\left(4-2\sqrt{3}\right)^2}}=\sqrt{4-2\sqrt{3}}=\sqrt{\left(\sqrt{3}-1\right)^2}=\sqrt{3}-1\)

Có -1 > -2 => \(\sqrt{3}-1>\sqrt{3}-2\Rightarrow\sqrt{\sqrt{28-16\sqrt{3}}}>\sqrt{3}-2\)