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Silly root question

  1. Nov 24, 2004 #1
    Here's a silly roots question that has my congested mind temporarily stumped:

    Let [tex]z = 1 + \sqrt{2}[/tex]. Find the five distinct fifth roots of z.

    Thanks in advance for helping me relieve the pressure.
     
  2. jcsd
  3. Nov 24, 2004 #2
    Some of the roots are going to be complex, so the way I would tackle the problem is to rewrite your number in the form:

    [tex]z=(1+\sqrt{2}){\rm e}^{2\pi ni}[/tex],

    where n=0,1,2,.... Then taking the fifth root gives:

    [tex]z^{1/5}=(1+\sqrt{2})^{1/5} {\rm e}^{2\pi ni/5}[/tex],

    which you can write in the form:

    [tex]z^{1/5}=(1+\sqrt{2})^{1/5} \left \{ \cos \left ( \frac{2\pi n}{5} \right) +i \sin \left( \frac{2\pi n}{5} \right) \right \}[/tex].

    Evaluating this for different n, should give 5 distinct roots.
     
  4. Nov 24, 2004 #3
    thank you very much for the insight...I now proceed to kick myself for not seeing it on my own {sound of kicking}

    Thanks!
     
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