Question Number 202383 by MATHEMATICSAM last updated on 25/Dec/23
$$\mathrm{Show}\:\mathrm{that}\:\frac{{a}\sqrt{{b}}\:−\:{b}\sqrt{{a}}}{{a}\sqrt{{b}}\:+\:{b}\sqrt{{a}}}\:=\:\frac{\mathrm{1}}{{a}\:−\:{b}}\left({a}\:+\:{b}\:−\:\mathrm{2}\sqrt{{ab}}\right). \\ $$
Answered by AST last updated on 25/Dec/23
$$=\frac{\left({a}\sqrt{{b}}−{b}\sqrt{{a}}\right)^{\mathrm{2}} }{{ab}\left({a}−{b}\right)}\:=\frac{{a}^{\mathrm{2}} {b}+{b}^{\mathrm{2}} {a}−\mathrm{2}{ab}\sqrt{{ab}}}{{ab}\left({a}−{b}\right)}=\frac{{a}+{b}−\mathrm{2}\sqrt{{ab}}}{{a}−{b}} \\ $$
Answered by lorance last updated on 25/Dec/23
$$\frac{{a}\sqrt{{b}}\:−\:{b}\sqrt{{a}}}{{a}\sqrt{{b}}\:+\:{b}\sqrt{{a}}}\:=\:\frac{\cancel{\sqrt{{ab}}}×\left(\sqrt{{a}}−\sqrt{{b}}\right)}{\:\cancel{\sqrt{{ab}}}×\left(\sqrt{{a}}+\sqrt{{b}}\right)}×\frac{\sqrt{{a}}−\sqrt{{b}}}{\:\sqrt{{a}}−\sqrt{{b}}}=\frac{\left(\sqrt{{a}}−\sqrt{{b}}\right)^{\mathrm{2}} }{\:\left(\sqrt{{a}}\right)^{\mathrm{2}} −\left(\sqrt{{b}}\right)^{\mathrm{2}} }=\frac{\mathrm{1}}{{a}−{b}}×\left({a}+{b}−\mathrm{2}\sqrt{{ab}}\right) \\ $$