Question

$$\begin{gathered} Solvethis: \\ \frac{1}{\sqrt{3}+\sqrt{2}}-\frac{2}{\sqrt{5}-\sqrt{3}}-\frac{3}{\sqrt{2}-\sqrt{5}} \end{gathered}$$

Answer 1

$$\begin{gathered} \frac{1}{\sqrt{3}+\sqrt{2}}-\frac{2}{\sqrt{5}{\displaystyle -}{\displaystyle \sqrt{3}}}-\frac{3}{\sqrt{2}{\displaystyle -}{\displaystyle \sqrt{5}}} \\ =\frac{{\displaystyle 1}}{{\displaystyle \sqrt{3}+\sqrt{2}}}\times \frac{\sqrt{3}{\displaystyle -}\sqrt{2}}{\sqrt{3}{\displaystyle -}\sqrt{2}}-\frac{{\displaystyle 2}}{{\displaystyle \sqrt{5}-\sqrt{3}}}\times \frac{\sqrt{5}{\displaystyle +}\sqrt{3}}{\sqrt{5}{\displaystyle +}\sqrt{3}}-\frac{3}{\sqrt{2}{\displaystyle -}{\displaystyle \sqrt{5}}}\times \frac{\sqrt{2}{\displaystyle +}\sqrt{5}}{\sqrt{2}{\displaystyle +}\sqrt{5}} \\ =\frac{\sqrt{3}-\sqrt{2}}{{\displaystyle {\left(\sqrt{3}\right)}^2}{\displaystyle -}{\displaystyle {{\displaystyle \left(\sqrt{2}\right)}}^2}}-\frac{2\left(\sqrt{5}+\sqrt{3}\right)}{{\displaystyle {\left(\sqrt{5}\right)}^2}{\displaystyle -}{\displaystyle {{\displaystyle \left(\sqrt{3}\right)}}^2}}-\frac{3\left(\sqrt{2}+\sqrt{5}\right)}{{\displaystyle {{\displaystyle \left(\sqrt{2}\right)}}^2}{\displaystyle -}{\left(\sqrt{5}\right)}^2} \\ =\frac{{\displaystyle \sqrt{3}-\sqrt{2}}}{{\displaystyle 3-2}}-\frac{2\left(\sqrt{5}+\sqrt{3}\right)}{{\displaystyle 5-3}}-\frac{3\left(\sqrt{2}+\sqrt{5}\right)}{{\displaystyle 2-5}} \\ =\frac{{\displaystyle \sqrt{3}-\sqrt{2}}}{{\displaystyle 1}}-\frac{2\left(\sqrt{5}+\sqrt{3}\right)}{{\displaystyle 2}}-\frac{3\left(\sqrt{2}+\sqrt{5}\right)}{{\displaystyle -3}} \\ =\sqrt{3}-\sqrt{2}-\sqrt{5}-\sqrt{3}+\sqrt{2}+\sqrt{5} \\ =0 \end{gathered}$$