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

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