Question Number 146325 by 7770 last updated on 12/Jul/21
$${Given}\:{that}\:\left({a}+{b}\right)=\sqrt{\mathrm{3}\sqrt{\mathrm{3}}−\sqrt{\mathrm{2}}} \\ $$$${and}\:\left({a}−{b}\right)=\sqrt{\mathrm{3}\sqrt{\mathrm{2}}−\sqrt{\mathrm{3}}} \\ $$$${Find} \\ $$$$\left({i}\right)\:{ab}\:\:\:\:\:\:\left({ii}\right)\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} \\ $$
Answered by Ar Brandon last updated on 12/Jul/21
$$\left(\mathrm{a}−\mathrm{b}\right)^{\mathrm{2}} =\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} −\mathrm{2ab} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\left(\mathrm{a}+\mathrm{b}\right)^{\mathrm{2}} −\mathrm{4ab} \\ $$$$\Rightarrow\mathrm{4ab}=\left(\mathrm{a}+\mathrm{b}\right)^{\mathrm{2}} −\left(\mathrm{a}−\mathrm{b}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{4}\left(\sqrt{\mathrm{3}}−\sqrt{\mathrm{2}}\right)\Rightarrow\mathrm{ab}=\sqrt{\mathrm{3}}−\sqrt{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} =\left(\mathrm{a}+\mathrm{b}\right)^{\mathrm{2}} −\mathrm{2ab}=\sqrt{\mathrm{3}}+\sqrt{\mathrm{2}} \\ $$