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Question Number 155571 by peter frank last updated on 02/Oct/21

Find the cube root of one .Hence  show that the sum of the root is   equal to zero

Findthecuberootofone.Henceshowthatthesumoftherootisequaltozero

Answered by JDamian last updated on 02/Oct/21

z^3 =1  {z}={e^(i((2π)/3)k) }_(0≤k<3)   Σ_(k=0) ^2 e^(i((2π)/3)k) =Σ_(k=0) ^2 (e^(i((2π)/3)) )^k =(((e^(i((2π)/3)) )^3 −1)/(e^(i((2π)/3)) −1))=((e^(i2π) −1)/(e^(i((2π)/3)) −1))=  =((1−1)/(e^(i((2π)/3)) −1))=0

z3=1{z}={ei2π3k}0k<32k=0ei2π3k=2k=0(ei2π3)k=(ei2π3)31ei2π31=ei2π1ei2π31==11ei2π31=0

Commented by peter frank last updated on 02/Oct/21

great sir ;thanks

greatsir;thanks

Commented by puissant last updated on 02/Oct/21

Nice sir.. but it is rather k∈{0,1,2}..

Nicesir..butitisratherk{0,1,2}..

Answered by puissant last updated on 02/Oct/21

z^3 =1 ⇒ z=e^(i((2kπ)/3))  ;  k∈[∣ 0;2 ∣]  → k=0 ⇒ z= 1  → k=1 ⇒ z=e^(i((2π)/3))   →k=2 ⇒ z=e^(i((4π)/3)) = e^(−i((2π)/3))   let j=e^(i((2π)/3))   1+j+j^2 = ((1−j^3 )/(1−j))= (((e^(i((2π)/3)) )^3 −1)/(e^(i((2π)/3)) −1))= 0...  −−−−−−−−−−−−−  j=e^(i((2π)/3))  = −(1/2)+((√3)/2)i and   j^2 =e^(i((4π)/3)) = e^(−i((2π)/3)) = −(1/2)−((√3)/2)i..  1+j+j^2 = 1−(1/2)+((√3)/2)i−(1/2)−((√3)/2)i=1−1=0  ⇒ 1+j+j^2 = 0...

z3=1z=ei2kπ3;k[0;2]k=0z=1k=1z=ei2π3k=2z=ei4π3=ei2π3letj=ei2π31+j+j2=1j31j=(ei2π3)31ei2π31=0...j=ei2π3=12+32iandj2=ei4π3=ei2π3=1232i..1+j+j2=112+32i1232i=11=01+j+j2=0...

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