Question Number 127078 by amns last updated on 26/Dec/20
$$\frac{\frac{\frac{\mathrm{1}}{\mathrm{2}}}{\:\sqrt{\mathrm{2}}}}{\frac{\frac{\frac{\mathrm{1}\:+\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}}{\mathrm{10}}}{\mathrm{3}}}{\mathrm{2}}}\:=\:? \\ $$
Commented by hknkrc46 last updated on 26/Dec/20
![((1/2) : (√2)) : [(1 + (1/( (√2)))) : 10 : 3 : 2] = ((1/2) ∙ (1/( (√2)))) : [(((√2) + 1)/( (√2))) ∙ (1/(10)) ∙ (1/3) ∙ (1/2)] = (1/2) ∙ (1/( (√2))) ∙ ((√2)/( (√2) + 1)) ∙ ((10)/1) ∙ (3/1) ∙ (2/1) = ((30)/( (√2) + 1)) = ((30((√2) − 1))/(((√2) + 1)((√2) − 1))) = 30(√2) − 30](https://www.tinkutara.com/question/Q127079.png)
$$\left(\frac{\mathrm{1}}{\mathrm{2}}\::\:\sqrt{\mathrm{2}}\right)\::\:\left[\left(\mathrm{1}\:+\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\right)\::\:\mathrm{10}\::\:\mathrm{3}\::\:\mathrm{2}\right] \\ $$$$=\:\left(\frac{\mathrm{1}}{\mathrm{2}}\:\centerdot\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\right)\::\:\left[\frac{\sqrt{\mathrm{2}}\:+\:\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:\centerdot\:\frac{\mathrm{1}}{\mathrm{10}}\:\centerdot\:\frac{\mathrm{1}}{\mathrm{3}}\:\centerdot\:\frac{\mathrm{1}}{\mathrm{2}}\right] \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{2}}\:\centerdot\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:\centerdot\:\frac{\sqrt{\mathrm{2}}}{\:\sqrt{\mathrm{2}}\:+\:\mathrm{1}}\:\centerdot\:\frac{\mathrm{10}}{\mathrm{1}}\:\centerdot\:\frac{\mathrm{3}}{\mathrm{1}}\:\centerdot\:\frac{\mathrm{2}}{\mathrm{1}} \\ $$$$=\:\frac{\mathrm{30}}{\:\sqrt{\mathrm{2}}\:+\:\mathrm{1}}\:=\:\frac{\mathrm{30}\left(\sqrt{\mathrm{2}}\:−\:\mathrm{1}\right)}{\left(\sqrt{\mathrm{2}}\:+\:\mathrm{1}\right)\left(\sqrt{\mathrm{2}}\:−\:\mathrm{1}\right)} \\ $$$$=\:\mathrm{30}\sqrt{\mathrm{2}}\:−\:\mathrm{30} \\ $$