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Prove-that-3-5-2-3-5-2-1-Question-216694-reposted-for-new-answers-

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Question Number 216739 by ArshadS last updated on 17/Feb/25
Prove that ^3 (√((√5)+2)) −^3 (√((√5)−2)) =1 Question#216694 reposted for new answers
Provethat35+2−35−2=1You can't use 'macro parameter character #' in math mode
Answered by Rasheed.Sindhi last updated on 17/Feb/25
(((√5) +2))^(1/3) −(((√5) −2))^(1/3) =1 let (((√5) +2))^(1/3) −(((√5) −2))^(1/3) =x And a=(√5) +2 , b=(√5) −2 ⇒a−b=4 , ab=1 a−b=((a)^(1/3) )^3 −(((b ))^(1/3) )^3 =((a)^(1/3) −(b)^(1/3) )(((a)^(1/3) )^2 +((b)^(1/3) )^2 +((a)^(1/3) )((b)^(1/3) )) =((a)^(1/3) −(b)^(1/3) )(((a)^(1/3) −(b)^(1/3) )^2 +3((a)^(1/3) )((b)^(1/3) )) 4=(x)( (x)^2 +3(1) ) x^3 +3x−4=0 (x−1)(x^2 +x+4)=0 x=1 or x^2 +x+4=0⇒x∉R proved
5+23−5−23=1let5+23−5−23=xAnda=5+2,b=5−2⇒a−b=4,ab=1a−b=(a3)3−(b3)3=(a3−b3)((a3)2+(b3)2+(a3)(b3))=(a3−b3)((a3−b3)2+3(a3)(b3))4=(x)((x)2+3(1))x3+3x−4=0(x−1)(x2+x+4)=0x=1orx2+x+4=0⇒x∉Rproved

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