Lời giải:
\(\frac{1}{1-\sqrt{2}}-\frac{1}{\sqrt{2}-\sqrt{3}}+\frac{1}{\sqrt{3}-\sqrt{4}}=\frac{1+\sqrt{2}}{(1-\sqrt{2})(1+\sqrt{2})}-\frac{\sqrt{2}+\sqrt{3}}{(\sqrt{2}-\sqrt{3})(\sqrt{2}+\sqrt{3})}+\frac{\sqrt{3}+\sqrt{4}}{(\sqrt{3}-\sqrt{4})(\sqrt{3}+\sqrt{4})}\)
\(=\frac{1+\sqrt{2}}{1-2}-\frac{\sqrt{2}+\sqrt{3}}{2-3}+\frac{\sqrt{3}+\sqrt{4}}{3-4}=-(1+\sqrt{2})+(\sqrt{2}+\sqrt{3})-(\sqrt{3}+\sqrt{4})\)
\(=-1-\sqrt{2}+\sqrt{2}+\sqrt{3}-\sqrt{3}-\sqrt{4}=-1-\sqrt{4}=-1-2=-3\)
\(\dfrac{1}{1-\sqrt{2}}-\dfrac{1}{\sqrt{2}-\sqrt{3}}+\dfrac{1}{\sqrt{3}-\sqrt{4}}\)
\(=\dfrac{\sqrt{2}+1}{\left(1-\sqrt{2}\right)\left(1+\sqrt{2}\right)}-\dfrac{\sqrt{2}+\sqrt{3}}{\left(\sqrt{2}-\sqrt{3}\right)\left(\sqrt{2}+\sqrt{3}\right)}+\dfrac{\sqrt{4}+\sqrt{3}}{\left(\sqrt{3}-\sqrt{4}\right)\left(\sqrt{3}+\sqrt{4}\right)}\)
\(=\dfrac{\sqrt{2}+1}{-1}-\dfrac{\sqrt{2}+\sqrt{3}}{-1}+\dfrac{\sqrt{4}+\sqrt{3}}{-1}=-1-\sqrt{2}+\sqrt{2}+\sqrt{3}-\sqrt{4}-\sqrt{3}\)
\(=-1-\sqrt{4}=-1-2=-3\)
\(\dfrac{1}{1-\sqrt{2}}-\dfrac{1}{\sqrt{2}-\sqrt{3}}+\dfrac{1}{\sqrt{3}-\sqrt{4}}\)
\(=-\sqrt{2}-1+\sqrt{3}+\sqrt{2}-2-\sqrt{3}\)
=-3
