Question Number 200169 by hardmath last updated on 15/Nov/23
$$\mathrm{Rationalise}\:\mathrm{the}\:\mathrm{deniminator}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{following}\:\mathrm{fraction}: \\ $$$$\frac{\mathrm{1}}{\:\sqrt{\mathrm{6}}\:−\:\sqrt{\mathrm{3}}\:+\:\sqrt{\mathrm{2}}\:+\:\mathrm{1}}\:=\:? \\ $$
Answered by Sutrisno last updated on 15/Nov/23
$$\frac{\mathrm{1}}{\:\sqrt{\mathrm{6}}−\sqrt{\mathrm{3}}+\sqrt{\mathrm{2}}+\mathrm{1}}×\frac{\left(\sqrt{\mathrm{6}}−\sqrt{\mathrm{3}}\right)−\left(\sqrt{\mathrm{2}}+\mathrm{1}\right)}{\left(\sqrt{\mathrm{6}}−\sqrt{\mathrm{3}}\right)−\left(\sqrt{\mathrm{2}}+\mathrm{1}\right)} \\ $$$$\frac{\left(\sqrt{\mathrm{6}}−\sqrt{\mathrm{3}}\right)−\left(\sqrt{\mathrm{2}}+\mathrm{1}\right)}{\left(\mathrm{6}−\mathrm{2}\sqrt{\mathrm{18}}+\mathrm{3}\right)−\left(\mathrm{2}+\mathrm{2}\sqrt{\mathrm{2}}+\mathrm{1}\right)} \\ $$$$\frac{\left(\sqrt{\mathrm{6}}−\sqrt{\mathrm{3}}\right)−\left(\sqrt{\mathrm{2}}+\mathrm{1}\right)}{\mathrm{6}−\mathrm{8}\sqrt{\mathrm{2}}}×\frac{\mathrm{6}+\mathrm{8}\sqrt{\mathrm{2}}}{\mathrm{6}+\mathrm{8}\sqrt{\mathrm{2}}} \\ $$$$\frac{\left(\left(\sqrt{\mathrm{6}}−\sqrt{\mathrm{3}}\right)−\left(\sqrt{\mathrm{2}}+\mathrm{1}\right)\right)\left(\mathrm{6}+\mathrm{8}\sqrt{\mathrm{2}}\right)}{−\mathrm{92}} \\ $$
Commented by Frix last updated on 15/Nov/23
$$=\frac{\mathrm{6}\sqrt{\mathrm{6}}−\mathrm{11}\sqrt{\mathrm{3}}+\mathrm{14}\sqrt{\mathrm{2}}+\mathrm{22}}{\mathrm{92}} \\ $$