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Question Number 208441 by efronzo1 last updated on 16/Jun/24

Answered by Etimbuk last updated on 16/Jun/24

x^3 −y^3 =0  x^(3 ) = y^3  ⇒ x = y            the stated fraction will be  ((x^(2016)  + (x^2 )^(1008)  + y^(2016) )/((2x)^(2016) )) = ((2x^(2016) + x^(2016) )/(2^(2016) x^(2016) ))  ((3x^(2016) )/(2^(2016) x^(2016) )) = (3/2^(2016) )

x3y3=0x3=y3x=ythestatedfractionwillbex2016+(x2)1008+y2016(2x)2016=2x2016+x201622016x20163x201622016x2016=322016

Commented by efronzo1 last updated on 16/Jun/24

Answered by Berbere last updated on 16/Jun/24

if y=0⇒x=0  y≠0 ⇒((x/y))^2 +((x/y))+1=0z=(x/y)  ⇒;z≠1z^2 +z+1=0⇔z^3 =1;z+1=−z^2   ((z^(2016) +z^(1008) +1)/((1+z)^(2016) ))=(((z^(3)^(672) ) +(z^3 )^(336) +1)/((−z^2 )^(2016) ))=(3/1)=3

ify=0x=0y0(xy)2+(xy)+1=0z=xy;z1z2+z+1=0z3=1;z+1=z2z2016+z1008+1(1+z)2016=(z3)672+(z3)336+1(z2)2016=31=3

Answered by Frix last updated on 16/Jun/24

x^2 +xy+y^2 =0  y=px∧x≠0 ⇒l  p^2 +p+1=0  ((p^(2016) +p^(1008) +1)/((p+1)^(2016) ))  p=e^(±i((2π)/3))  ⇒ p+1=e^(±i(π/3))   p^(3n) =1∧(p+1)^(6n) =1 ⇒ ((p^(2016) +p^(1008) +1)/((p+1)^(2016) ))=3

x2+xy+y2=0y=pxx0lp2+p+1=0p2016+p1008+1(p+1)2016p=e±i2π3p+1=e±iπ3p3n=1(p+1)6n=1p2016+p1008+1(p+1)2016=3

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