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If z is a complex number satisfying z + 1/z =1, calculate z^12+ 1/z^12.

 Jan 15, 2021
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(z + 1/z)^2   = 1

z^2  + 2z (1/z) + 1/z^2  = 1

z^2 + 2 + 1/z^2  =  1                 subtract 2 from both sides

z^2 + 1/z^2  =  -1

 

(z^2 + 1/z^2)^2  =  1

z^4 + 2 z^2(1/z^2) + 1/z^4  =  1

z^4 + 2 + 1/z^4  =  1               subtract 2 from  both sides

z^4 + 1/z^4  =  -1

 

(z^4 + 1/z^4)^3    = ( -1 )^3            binomial expansion on the left side

 

(z^4)^3  + 3(z^4)^2(1/z^4)  + 3(z^4)(1/z^4)^2  + (1/z^4)^3  =  -1

 

z^12  + 3(z^8/z^4) + 3(z^4/z^8 + 1/z^12   =  -1

 

z^12 + 3z^4 + 3/z^4  + 1/z^12  =  -1

 

z^12 + 3 (z^4 + 1/z^4)  + 1/z^12  =  -1

 

z^12 + 3(-1) +1/z^12  =  -1

 

z^12 + 1/z^12  - 3  =  -1          add 3 to  both sides

 

z^12 + 1/z^12  =  2

 

 

cool cool cool

 Jan 15, 2021

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