Chứng minh rằng \(\sqrt{5}-\sqrt{3}>\sqrt{3}-\sqrt{2}\)
Hướng dẫn giải
Bình phương ta có: \(\left(\sqrt{5}-\sqrt{3}\right)^2=5+3-2\sqrt{5}.\sqrt{3}=8-2\sqrt{5.3}\)
\(=8-2\sqrt{15}=5-\left(2\sqrt{15}-3\right)\)
\(\left(\sqrt{3}-\sqrt{2}\right)^2=5-2\sqrt{3}.\sqrt{2}=5-2\sqrt{6}\)
Tiếp tục so sánh \(2\sqrt{6}\) và \(2\sqrt{15}-3\)
\(\left(2\sqrt{6}\right)^2=4.6=24.\)
\(\left(2\sqrt{15}-3\right)^2=69-12\sqrt{15}=24+45-12\sqrt{15}\)
\(45^2=2025< \left(12\sqrt{15}\right)^2=2160\) nên \(2\sqrt{15}-3< 2\sqrt{6}\)
Vậy thì \(\left(\sqrt{5}-\sqrt{3}\right)^2>\left(\sqrt{3}-\sqrt{2}\right)^2\) hay \(\sqrt{5}-\sqrt{3}>\sqrt{3}-\sqrt{2}\).