Question Number 122174 by AbdullahMohammadNurusSafa last updated on 14/Nov/20
$${If}\:\:\boldsymbol{{x}}\:=\:\sqrt{\mathrm{5}\:}\:+\:\sqrt{\mathrm{3}},{then}\:\boldsymbol{{x}}^{\mathrm{3}} \:+\:\frac{\mathrm{1}}{\boldsymbol{{x}}^{\mathrm{3}} }\:=\:? \\ $$$$\boldsymbol{{or}},\:\boldsymbol{{is}}\:\boldsymbol{{it}}\:\boldsymbol{{possible}}\:\boldsymbol{{at}}\:\boldsymbol{{all}}? \\ $$
Answered by [email protected] last updated on 14/Nov/20
$$\mathrm{x}^{\mathrm{3}} +\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{3}} }=\left(\mathrm{x}+\frac{\mathrm{1}}{\mathrm{x}}\right)^{\mathrm{3}} −\mathrm{3}\left(\mathrm{x}+\frac{\mathrm{1}}{\mathrm{x}}\right)= \\ $$$$=\left(\mathrm{5}\sqrt{\mathrm{5}}+\mathrm{3}\sqrt{\mathrm{3}}+\mathrm{15}\sqrt{\mathrm{3}}+\mathrm{27}\sqrt{\mathrm{5}}\right)−\mathrm{3}\left(\sqrt{\mathrm{5}}+\sqrt{\mathrm{3}}\right)= \\ $$$$=\mathrm{29}\sqrt{\mathrm{5}}+\mathrm{15}\sqrt{\mathrm{3}}\:. \\ $$