Question Number 226362 by Rojarani last updated on 26/Nov/25
Answered by Frix last updated on 26/Nov/25
$$\alpha=\sqrt{\mathrm{3}}\mathrm{e}^{−\mathrm{i}\frac{\mathrm{3}\pi}{\mathrm{4}}} \\ $$$$\beta=\sqrt{\mathrm{3}}\mathrm{e}^{\mathrm{i}\frac{\mathrm{3}\pi}{\mathrm{4}}} \\ $$$$\alpha^{{n}} +\beta^{{n}} =\mathrm{2}×\mathrm{3}^{\frac{{n}}{\mathrm{2}}} \mathrm{cos}\:\frac{\mathrm{3}{n}\pi}{\mathrm{4}} \\ $$$$\frac{\left(\alpha^{\mathrm{23}} +\beta^{\mathrm{23}} \right)+\left(\alpha^{\mathrm{14}} +\beta^{\mathrm{14}} \right)}{\left(\alpha^{\mathrm{15}} +\beta^{\mathrm{15}} \right)+\left(\alpha^{\mathrm{10}} +\beta^{\mathrm{10}} \right)}=\frac{−\mathrm{177147}\sqrt{\mathrm{6}}+\mathrm{0}}{−\mathrm{2187}\sqrt{\mathrm{6}}+\mathrm{0}}= \\ $$$$=\mathrm{81} \\ $$