Question Number 157007 by Armindo last updated on 18/Oct/21
Commented by Armindo last updated on 18/Oct/21
I need help, for to solve This exercice.
Answered by TheHoneyCat last updated on 21/Oct/21
$$\frac{\mathrm{1}}{\:^{\mathrm{4}} \sqrt{\mathrm{2}}+^{\mathrm{4}} \sqrt{\mathrm{4}}+^{\mathrm{4}} \sqrt{\mathrm{8}}+\mathrm{2}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{1}/\mathrm{4}} +\mathrm{4}^{\mathrm{1}/\mathrm{4}} +\mathrm{8}^{\mathrm{1}/\mathrm{4}} +\mathrm{2}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{1}/\mathrm{4}} +\mathrm{2}^{\mathrm{2}/\mathrm{4}} +\mathrm{2}^{\mathrm{3}/\mathrm{4}} +\mathrm{2}^{\mathrm{4}/\mathrm{4}} } \\ $$$$=\left(\underset{{k}=\mathrm{1}} {\overset{\mathrm{4}} {\sum}}\left(\mathrm{2}^{\mathrm{1}/\mathrm{4}} \right)^{{k}} \right)^{−\mathrm{1}} \\ $$$$=\left(\mathrm{2}^{\mathrm{1}/\mathrm{4}} \frac{\mathrm{2}^{\mathrm{4}/\mathrm{4}} −\mathrm{1}}{\mathrm{2}^{\mathrm{1}/\mathrm{4}} −\mathrm{1}}\right)^{−\mathrm{1}} =\frac{\sqrt[{\mathrm{4}}]{\mathrm{2}}−\mathrm{1}}{\:\sqrt[{\mathrm{4}}]{\mathrm{2}}}\approx\mathrm{0}.\mathrm{2} \\ $$$$ \\ $$$${the}\:{second}\:{formula}\:{has}\:{no}\:{simplification}... \\ $$