Question Number 200729 by mnjuly1970 last updated on 22/Nov/23
$$ \\ $$$$\:\:\:{if}\:\:\:\:{a}<{b}<\mathrm{0}\:\:\: \\ $$$$\:\:\:{and}\:\:\sqrt{\:{a}^{\mathrm{4}} +\mathrm{2}{a}^{\mathrm{2}} {b}^{\mathrm{2}} +{b}^{\mathrm{4}} }\:+\sqrt{{a}^{\mathrm{4}} −\mathrm{2}{a}^{\mathrm{2}} {b}^{\mathrm{2}} +{b}^{\mathrm{4}} }\:=\:? \\ $$$$ \\ $$$$\:\:\: \\ $$$$\:\:\:\:\:\: \\ $$
Answered by AST last updated on 22/Nov/23
$$\sqrt{\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)^{\mathrm{2}} }+\sqrt{\left({a}^{\mathrm{2}} −{b}^{\mathrm{2}} \right)^{\mathrm{2}} }=\mid{a}^{\mathrm{2}} +{b}^{\mathrm{2}} \mid+\mid{a}^{\mathrm{2}} −{b}^{\mathrm{2}} \mid \\ $$$$={a}^{\mathrm{2}} +{b}^{\mathrm{2}} +\left({a}^{\mathrm{2}} −{b}^{\mathrm{2}} \right)=\mathrm{2}{a}^{\mathrm{2}} \\ $$