比较 $\frac{1}{\sqrt{6}-\sqrt{5}}$ 与 $\frac{1}{\sqrt{7}-\sqrt{6}}$ 的大小

答案

解： $\frac{1}{\sqrt{6}-\sqrt{5}}=\frac{\sqrt{6}+\sqrt{5}}{(\sqrt{6}-\sqrt{5})(\sqrt{6}+\sqrt{5})}=\sqrt{6}+\sqrt{5}$
$\frac{1}{\sqrt{7}-\sqrt{6}}=\frac{\sqrt{7}+\sqrt{6}}{(\sqrt{7}-\sqrt{6})(\sqrt{7}+\sqrt{6})}=\sqrt{7}+\sqrt{6}$
∵ $\sqrt{6}+\sqrt{5}<\sqrt{7}+\sqrt{6}$
∴ $\frac{1}{\sqrt{6}-\sqrt{5}}<\frac{1}{\sqrt{7}-\sqrt{6}}$