Question Number 190027 by Shrinava last updated on 26/Mar/23
$$\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{b}^{\mathrm{2}} \:=\:\mathrm{12} \\ $$$$\mathrm{ab}\:=\:\mathrm{4} \\ $$$$\mathrm{Fund}:\:\:\:\mathrm{a}^{\mathrm{3}} \:+\:\mathrm{b}^{\mathrm{3}} \:=\:? \\ $$
Answered by SEKRET last updated on 26/Mar/23
$$\:_{} +\:\begin{cases}{\boldsymbol{{a}}^{\mathrm{2}} +\boldsymbol{{b}}^{\mathrm{2}} =\mathrm{12}}\\{\mathrm{2}\boldsymbol{{ab}}=\mathrm{8}}\end{cases} \\ $$$$\:\:\:\:\:\:\:\left(\boldsymbol{{a}}+\boldsymbol{{b}}\right)^{\mathrm{2}} =\mathrm{20} \\ $$$$\:\:\:\:\:\boldsymbol{{a}}+\boldsymbol{{b}}\:=\:\mp\mathrm{2}\sqrt{\mathrm{5}} \\ $$$$\:\:\:\boldsymbol{\mathrm{a}}^{\mathrm{3}} +\boldsymbol{\mathrm{b}}^{\mathrm{3}} +\mathrm{3}\boldsymbol{\mathrm{ab}}\left(\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{b}}\right)\:=\:\left(\mp\mathrm{2}\sqrt{\mathrm{5}}\:\right)^{\mathrm{3}} \\ $$$$\:\:\:\:\boldsymbol{\mathrm{a}}^{\mathrm{3}} +\boldsymbol{\mathrm{b}}^{\mathrm{3}} =\:−\mathrm{12}\centerdot\left(\mp\mathrm{2}\sqrt{\mathrm{5}}\:\right)\:\mp\left(\mathrm{2}\sqrt{\mathrm{5}}\:\right)^{\mathrm{3}} \\ $$$$\:\:\:\:\boldsymbol{\mathrm{a}}^{\mathrm{3}} +\boldsymbol{\mathrm{b}}^{\mathrm{3}} =\mp\mathrm{16}\sqrt{\mathrm{5}} \\ $$