Question

So sánh

\(\sqrt{4+\sqrt{7}-}\sqrt{4-\sqrt{7}}va\sqrt{3}\)

Answer 1

Ta có:

\(\sqrt{4+\sqrt{7}}-\sqrt{4-\sqrt{7}}\) và \(\sqrt{3}\)

Nhân \(\sqrt{2}\) vào 2 vế ta có:

VT=\(\sqrt{8+2\sqrt{7}}-\sqrt{8-2\sqrt{7}}\)

VP=\(\sqrt{12}\)

Xét VT ta có:

\(VT=\sqrt{\left(\sqrt{7}+1\right)^2}-\sqrt{\left(\sqrt{7}-1\right)^2}\)

=\(\sqrt{7}+1-\left(\sqrt{7}-1\right)\)=1+1=2

Xét VP có:

\(\sqrt{12}=2\sqrt{3}\)

\(\Leftrightarrow VT< VP\) hay \(\sqrt{4+\sqrt{7}}-\sqrt{4-\sqrt{7}}< \sqrt{12}\)