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Question Number 95920 by bobhans last updated on 28/May/20

((54+(√x)))^(1/(3 )) + ((54−(√x)))^(1/(3 )) = ((18))^(1/(3 )) x = ?

54+x3+54−x3=183x=?

Answered by i jagooll last updated on 28/May/20

let a = ((54+(√x)))^(1/(3 )) , b = ((54−(√x)))^(1/(3 )) a+b = ((a^3 +b^3 )/(a^2 −ab+b^2 )) a+b = ((54)/((((54+(√x))^2 ))^(1/(3 )) −((54^2 −x))^(1/(3 )) +((54−(√x)))^(1/(3 )) )) ((18 ))^(1/(3 )) ( (((54+(√x))^2 ))^(1/(3 )) −((54^2 −x))^(1/(3 )) +((54−(√x))^2 ))^(1/(3 )) )=54

leta=54+x3,b=54−x3a+b=a3+b3a2−ab+b2a+b=54(54+x)23−542−x3+54−x3183((54+x)23−542−x3+54−x)23)=54

Answered by MJS last updated on 28/May/20

use this “trick”: a+b=c (a+b)^3 =c^3 a^3 +3a^2 b+3ab^2 +b^3 =c^3 3ab(a+b)=c^3 −a^3 −b^3 [a+b=c] 3abc=c^3 −a^3 −b^3 27a^3 b^3 c^3 =(c^3 −a^3 −b^3 )^3 in our case a=((α+β))^(1/3) ; b=((α−β))^(1/3) ; c=((18))^(1/3) 27(α+β)(α−β)18=(18−(α+β)−(α−β))^3 486(α^2 −β^2 )=(18−2α)^3 with α=54; β=(√x) 486(2916−x)=−729000 x=4416

usethis‘‘trick″:a+b=c(a+b)3=c3a3+3a2b+3ab2+b3=c33ab(a+b)=c3−a3−b3[a+b=c]3abc=c3−a3−b327a3b3c3=(c3−a3−b3)3inourcasea=α+β3;b=α−β3;c=18327(α+β)(α−β)18=(18−(α+β)−(α−β))3486(α2−β2)=(18−2α)3withα=54;β=x486(2916−x)=−729000x=4416

Commented by i jagooll last updated on 29/May/20

thank you sir

thankyousir

Answered by behi83417@gmail.com last updated on 28/May/20

54+(√x)=a^3 ,54−(√x)=b^3 ,((18))^(1/3) =c ⇒ { ((a+b=((18))^(1/3) )),((a^3 +b^3 =108)) :} ⇒(a+b)([(a+b)^2 −3ab]=108 ⇒((18))^(1/3) .[(((18))^(1/3) )^2 −3ab]=108 ⇒18−3abc=108⇒ab=((108−18)/(−3((18))^(1/3) ))=−((30)/((18))^(1/3) ) ⇒a^3 .b^3 =−((27000)/(18))=−1500 { ((a^3 +b^3 =108)),((a^3 .b^3 =−1500)) :}⇒z^2 −108z−1500=0 ⇒z=[a^3 ∨b^3 ]=54±(√(54^2 +1500))=54±(√(4416)) ⇒54±(√x)=54±(√(4416))⇒x=4416 ■.

54+x=a3,54−x=b3,183=c⇒{a+b=183a3+b3=108⇒(a+b)([(a+b)2−3ab]=108⇒183.[(183)2−3ab]=108⇒18−3abc=108⇒ab=108−18−3183=−30183⇒a3.b3=−2700018=−1500{a3+b3=108a3.b3=−1500⇒z2−108z−1500=0⇒z=[a3∨b3]=54±542+1500=54±4416⇒54±x=54±4416⇒x=4416◼.

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