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if-a-lt-b-lt-0-and-a-4-2a-2-b-2-b-4-a-4-2a-2-b-2-b-4-




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

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