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Question Number 153505 by mathdanisur last updated on 07/Sep/21

Commented by MJS_new last updated on 08/Sep/21

((x^2 +y^2 +5))^(1/3) −(√(x^2 +y^2 −5))=5 has no solution ∈C

x2+y2+53−x2+y2−5=5hasnosolution∈C

Answered by liberty last updated on 08/Sep/21

let x^2 +y^2 =u ⇒((u+5))^(1/3) −(√(u−5)) = 5 let ((u+5))^(1/3) =v⇒u=v^3 −5 ⇒v−5=(√(v^3 −10)) ⇒v^2 −10v+25=v^3 −10 ⇒v^3 −v^2 +10v−35=0 Let λ=((−25+x^4 +y^4 +2x^2 y^2 ))^(1/3) λ=(((x^2 +y^2 )^2 −25))^(1/3) =((u^2 −25))^(1/3) λ=(((u−5)(u+5)))^(1/3)

letx2+y2=u⇒u+53−u−5=5letu+53=v⇒u=v3−5⇒v−5=v3−10⇒v2−10v+25=v3−10⇒v3−v2+10v−35=0Letλ=−25+x4+y4+2x2y23λ=(x2+y2)2−253=u2−253λ=(u−5)(u+5)3

Commented by mr W last updated on 08/Sep/21

((u+5))^(1/3) −(√(u−5)) = 5 has no solution!

u+53−u−5=5hasnosolution!

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