Question

So sánh:

a) x=\(\sqrt{4+\sqrt{7}}-\sqrt{4-\sqrt{7}}\) và y=\(\sqrt{2}\)

b) x=\(\sqrt{\sqrt{6+\sqrt{20}}}\) và y=\(\sqrt{1+\sqrt{6}}\)

Answer 1

\(\text{a) }Ta\text{ }có:x^2=\left(\sqrt{4+\sqrt{7}}-\sqrt{4-\sqrt{7}}\right)^2\\ =4+\sqrt{7}-2\sqrt{\left(4+\sqrt{7}\right)\left(4-\sqrt{7}\right)}+4-\sqrt{7}\\ =8-2\sqrt{16-7}=8-6=2\\ \Rightarrow x=\sqrt{2}=y\)

\(\text{b) }Ta\text{ }có:\text{ }x=\sqrt{\sqrt{6+\sqrt{20}}}=\sqrt{\sqrt{6+2\sqrt{5}}}\\ =\sqrt{\sqrt{5+1+2\sqrt{5}}}=\sqrt{\sqrt{\left(\sqrt{5}+1\right)^2}}\\ =\sqrt{\sqrt{5}+1}< \sqrt{\sqrt{6}+1}=y\)