SOLUTION: If r is the complex root of {{{z^5 = -1}}} with smallest pos. argument, evaluate: {{{(1-r+r^2+r^4)^5}}}

Question 1167497: If r is the complex root of with smallest pos. argument, evaluate:

Answer by ikleyn(52878) (Show Source): You can put this solution on YOUR website!
.


There is this identity

     = 


which is valid for any real or complex number z.



    (This formula is the same as the formula for the sum of first 5 terms of a geometric progression).



So, if z is the root of the equation   = 0  (any of five complex roots), then

     = 0.    (1)



Thus in our case

     = 0,    (2)


where r is the complex root of  with smallest positive argument.



    Since in our case the factor  r+1 is not zero  (because r =/= -1).
    we can divide both sides of (2) by (r+1).



We will get then

         = 0.



It implies that

         = 0.      ANSWER

Solved, answered and explained.

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There is a bunch of my lessons on complex numbers
    - Complex numbers and arithmetical operations on them
    - Complex plane
    - Addition and subtraction of complex numbers in complex plane
    - Multiplication and division of complex numbers in complex plane
    - Raising a complex number to an integer power
    - How to take a root of a complex number
    - Solution of the quadratic equation with real coefficients on complex domain
    - How to take a square root of a complex number
    - Solution of the quadratic equation with complex coefficients on complex domain

    - Solved problems on taking roots of complex numbers
    - Solved problems on arithmetic operations on complex numbers
    - Solved problem on taking square root of complex number
    - Miscellaneous problems on complex numbers
    - Advanced problem on complex numbers
    - Solved problems on de'Moivre formula
    - Calculating the sum 1*sin(1°) + 2*sin(2°) + 3*sin(3°) + . . . + 180*sin(180°)
    - A curious example of an equation in complex numbers which HAS NO a solution
    - Solving one non-standard equation in complex numbers
in this site.

Also, you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic "Complex numbers".

Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.

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