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Hints on solving this equation

  1. Apr 9, 2009 #1
    1. The problem statement, all variables and given/known data

    The solutions of the equation a^3 z^3+b^3 i =0
    where a,b are an element of R+


    3. The attempt at a solution
    a^3 z^3 = - b^3 i
    z^3 = (-b^3 i) / (a^3)

    Now I'm stuck.
    Any other suggestions?
     
  2. jcsd
  3. Apr 9, 2009 #2

    statdad

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    Homework Helper

    Note that

    [tex]
    i^3 = -i
    [/tex]

    As a start you can rewrite the entire right side as a 3rd power.
     
  4. Apr 10, 2009 #3
    You may also want to write z in terms of its real and imaginary components.
     
  5. Apr 10, 2009 #4
    That has nothing to do with the question.
     
  6. Apr 10, 2009 #5
    Actually, it has. In fact, I think it is a genius way to solve the problem! Just making sure, we are asked to solve z in terms of a, b?
    Really, try his way, and see what you can do next.
    Because after doing that substitution, basically there is only one more step to find get the solution
     
  7. Apr 11, 2009 #6

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    It has everything to do with the equation. Assuming that a and b are real numbers, the cube root of [itex]-a^3/b^3[/itex] is -a/b so the only question is the cube root of i. Knowing that [itex]i^3= -i[/itex] helps with that. Warning: [itex]-ia^3/b^3[/itex]
    has three cube roots.
     
  8. Apr 11, 2009 #7

    arildno

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    Gold Member
    Dearly Missed

    Re-write the left-hand side as:
    [tex](az)^{3}-(ib)^{3}=0\to{(az-ib)((az)^{2}+iazb-b^{2})=0[/tex]
     
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