Question Number 224453 by Rojarani last updated on 13/Sep/25
Commented by fantastic last updated on 13/Sep/25
$$−\mathrm{144}? \\ $$
Commented by Rojarani last updated on 13/Sep/25
$$\:{Sir},\:{I}\:{don}'{t}\:{know}\:{the}\:{answer}. \\ $$
Commented by Ghisom_ last updated on 14/Sep/25
$$\mathrm{I}\:\mathrm{think}\:{a},\:{b},\:{c}\:\mathrm{are}\:\mathrm{the}\:\mathrm{solutions}\:\mathrm{of} \\ $$$${x}^{\mathrm{3}} −\mathrm{6}{x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{1}=\mathrm{0} \\ $$$$\Rightarrow\:{a}+{b}+{c}=\mathrm{6},\:{ab}+{bc}+{ca}=\mathrm{3},\:{abc}=−\mathrm{1} \\ $$$$\Rightarrow\:\mathrm{indeed}\:{ab}^{\mathrm{2}} +{bc}^{\mathrm{2}} +{ca}^{\mathrm{2}} =\mathrm{24} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}^{\mathrm{2}} {b}+{b}^{\mathrm{2}} {c}+{c}^{\mathrm{2}} {a}=−\mathrm{3} \\ $$$$ \\ $$$$\Rightarrow\:{a}^{−\mathrm{3}} ,\:{b}^{−\mathrm{3}} ,\:{c}^{−\mathrm{3}} \mathrm{are}\:\mathrm{the}\:\mathrm{solutions}\:\mathrm{of} \\ $$$${y}^{\mathrm{3}} +\mathrm{84}{y}^{\mathrm{2}} +\mathrm{159}{y}+\mathrm{1}=\mathrm{0} \\ $$$$\Rightarrow\:\mathrm{the}\:\mathrm{answer}\:\mathrm{should}\:\mathrm{be}\:−\mathrm{84} \\ $$
Commented by fantastic last updated on 14/Sep/25
$${I}\:{got} \\ $$$$\frac{\mathrm{1}}{{a}^{\mathrm{3}} }+\frac{\mathrm{1}}{{b}^{\mathrm{3}} }+\frac{\mathrm{1}}{{c}^{\mathrm{3}} }=\frac{\mathrm{144}\:+\mathrm{12}{abc}−\left({a}+{b}+{c}\right)^{\mathrm{3}} }{{a}^{\mathrm{2}} {b}^{\mathrm{2}} {c}^{\mathrm{2}} } \\ $$
Commented by Ghisom_ last updated on 14/Sep/25
$$\mathrm{insert}\:\mathrm{the}\:\mathrm{values}\:\mathrm{I}\:\mathrm{gave}\:\Rightarrow\:\mathrm{your}\:\mathrm{result}\:\mathrm{also} \\ $$$$\mathrm{is}\:−\mathrm{84} \\ $$
Commented by fantastic last updated on 14/Sep/25
$${thanks} \\ $$