Question Number 187989 by mnjuly1970 last updated on 24/Feb/23
Answered by HeferH last updated on 24/Feb/23
$$\:{b}^{\mathrm{2}} ={ac} \\ $$$$\:{b}^{\mathrm{3}} ={abc} \\ $$$$\:\frac{{b}^{\mathrm{3}} {c}^{\mathrm{3}} +{a}^{\mathrm{3}} {c}^{\mathrm{3}} +{a}^{\mathrm{3}} {b}^{\mathrm{3}} }{{a}^{\mathrm{3}} {b}^{\mathrm{3}} {c}^{\mathrm{3}} }\centerdot\left(\frac{{a}^{\mathrm{2}} {b}^{\mathrm{2}} {c}^{\mathrm{2}} }{{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} }\right)= \\ $$$$\:\frac{{b}^{\mathrm{3}} {c}^{\mathrm{3}} +{a}^{\mathrm{3}} {c}^{\mathrm{3}} +{a}^{\mathrm{3}} {b}^{\mathrm{3}} }{{abc}}\centerdot\left(\frac{\mathrm{1}}{{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} }\right)= \\ $$$$\:\frac{\left({bc}\right)^{\mathrm{3}} +{b}^{\mathrm{6}} +\left({ab}\right)^{\mathrm{3}} }{{b}^{\mathrm{3}} }\centerdot\left(\frac{\mathrm{1}}{{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} }\right)= \\ $$$$\:{c}^{\mathrm{3}} +\:{b}^{\mathrm{3}} \:+\:{a}^{\mathrm{3}} \centerdot\left(\frac{\mathrm{1}}{{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} }\right)=\:\mathrm{1} \\ $$$$\: \\ $$
Answered by MathGuy last updated on 24/Feb/23
$${Answer}\::−\: \\ $$$$\frac{{a}^{\mathrm{2}} {b}^{\mathrm{2}} {c}^{\mathrm{2}} }{{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} }\left(\frac{{b}^{\mathrm{3}} {c}^{\mathrm{3}} +{c}^{\mathrm{3}} {a}^{\mathrm{3}} +{a}^{\mathrm{3}} {b}^{\mathrm{3}} }{{a}^{\mathrm{3}} {b}^{\mathrm{3}} {c}^{\mathrm{3}} }\right) \\ $$$$ \\ $$$$\Rightarrow\:\frac{\mathrm{1}}{{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} }\left(\frac{{b}^{\mathrm{3}} {c}^{\mathrm{3}} +{c}^{\mathrm{3}} {a}^{\mathrm{3}} +{a}^{\mathrm{3}} {b}^{\mathrm{3}} }{{abc}}\right) \\ $$$$ \\ $$$$\Rightarrow\:\frac{\mathrm{1}}{{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} }\left(\frac{{b}^{\mathrm{2}} {c}^{\mathrm{2}} }{{a}}+\frac{{c}^{\mathrm{2}} {a}^{\mathrm{2}} }{{b}}+\frac{{a}^{\mathrm{2}} {b}^{\mathrm{2}} }{{c}}\right) \\ $$$${substitute}\:{b}^{\mathrm{2}} ={ac}\:\&\:{c}^{\mathrm{2}} {a}^{\mathrm{2}} ={b}^{\mathrm{2}} ,\:{in}\:{the}\:{expression} \\ $$$${under}\:{brackets}\:{and}\:{simplify}. \\ $$$${you}\:{will}\:{get}\: \\ $$$$\frac{\mathrm{1}}{{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} }\left({a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} \right) \\ $$$$=\:\mathrm{1}\: \\ $$$$ \\ $$$$ \\ $$