Question Number 174719 by AgniMath last updated on 09/Aug/22
Answered by behi834171 last updated on 09/Aug/22
$$\boldsymbol{{a}}+\boldsymbol{{b}}\neq\mathrm{0} \\ $$$$\Rightarrow\frac{\boldsymbol{{a}}+\boldsymbol{{b}}}{\boldsymbol{{ab}}}=\frac{\mathrm{1}}{\boldsymbol{{a}}+\boldsymbol{{b}}}\Rightarrow\left(\boldsymbol{{a}}+\boldsymbol{{b}}\right)^{\mathrm{2}} =\boldsymbol{{ab}}\Rightarrow \\ $$$$\Rightarrow\boldsymbol{{a}}^{\mathrm{2}} +\boldsymbol{{ab}}+\boldsymbol{{b}}^{\mathrm{2}} =\mathrm{0}\overset{\boldsymbol{{a}}\neq\boldsymbol{{b}}} {\Rightarrow}\left(\boldsymbol{{a}}−\boldsymbol{{b}}\right)\left(\boldsymbol{{a}}^{\mathrm{2}} +\boldsymbol{{ab}}+\boldsymbol{{b}}^{\mathrm{2}} \right)=\mathrm{0} \\ $$$$\Rightarrow\boldsymbol{{a}}^{\mathrm{3}} −\boldsymbol{{b}}^{\mathrm{3}} =\mathrm{0}\Rightarrow\boldsymbol{{a}}^{\mathrm{3}} =\boldsymbol{{b}}^{\mathrm{3}} \:\:\:\:.\blacksquare \\ $$