Question Number 190942 by mathlove last updated on 14/Apr/23
$$\left({x}+{y}\right)\left({x}^{\mathrm{2}} −{xy}+{y}^{\mathrm{2}} \right)=\mathrm{2} \\ $$$$\frac{{x}^{\mathrm{3}} {y}^{\mathrm{3}} }{{x}^{\mathrm{9}} +{y}^{\mathrm{9}} −\mathrm{8}}=? \\ $$
Answered by Frix last updated on 15/Apr/23
$${x}^{\mathrm{3}} +{y}^{\mathrm{3}} =\mathrm{2}\:\Rightarrow\:{y}^{\mathrm{3}} =\mathrm{2}−{x}^{\mathrm{3}} \\ $$$$\frac{{x}^{\mathrm{3}} \left(\mathrm{2}−{x}^{\mathrm{3}} \right)}{{x}^{\mathrm{9}} +\left(\mathrm{2}−{x}^{\mathrm{3}} \right)^{\mathrm{3}} −\mathrm{8}}=\frac{{x}^{\mathrm{3}} \left(\mathrm{2}−{x}^{\mathrm{3}} \right)}{−\mathrm{6}{x}^{\mathrm{3}} \left(\mathrm{2}−{x}^{\mathrm{3}} \right)}=−\frac{\mathrm{1}}{\mathrm{6}} \\ $$
Answered by manxsol last updated on 15/Apr/23
$${x}^{\mathrm{3}} +{y}^{\mathrm{3}} =\mathrm{2} \\ $$$$\left({x}^{\mathrm{3}} +{y}^{\mathrm{3}} \right)^{\mathrm{3}} ={x}^{\mathrm{9}} +{y}^{\mathrm{9}} +\mathrm{3}{x}^{\mathrm{3}} {y}^{\mathrm{3}} \left({x}^{\mathrm{3}} +{y}^{\mathrm{3}} \right) \\ $$$$\mathrm{8}={x}^{\mathrm{9}} +{y}^{\mathrm{9}} +\mathrm{6}{x}^{\mathrm{3}} {y}^{\mathrm{3}} \\ $$$$−\mathrm{6}{x}^{\mathrm{3}} {y}^{\mathrm{3}} ={x}^{\mathrm{9}} +{y}^{\mathrm{9}} −\mathrm{8} \\ $$$$\frac{{x}^{\mathrm{3}} {y}^{\mathrm{3}} }{−\mathrm{6}{x}^{\mathrm{3}} {y}^{\mathrm{3}} }=−\frac{\mathrm{1}}{\mathrm{6}} \\ $$