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A Number Raised to the m Power

  1. Feb 2, 2014 #1
    Hello Everyone,

    I was wondering, does anyone know of a proof that showed if a number is raised to the mth power, where m is a positive even number, the number is always real?
     
  2. jcsd
  3. Feb 2, 2014 #2

    Borek

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    Staff: Mentor

    (1+2i)4 = -7-24i
     
  4. Feb 2, 2014 #3
    Drats! I was hoping it was true! How about if m were only 2? Would the statement then be true?
     
  5. Feb 2, 2014 #4
    It isn't: (a + bi)(a + bi) = a^2 + 2abi + b^2i^2 = a^2 + 2abi - b^2
     
  6. Feb 2, 2014 #5
    ##(1+i)^2=1+2i-1=2i##

    Your claim will only be true for ##m=2## if the number is only real or imaginary. This is clear by expanding a complex number as a binomial.

    ##(a+bi)^2=a^2-b^2+2abi##

    In order for this to be real, either ##a## or ##b## must be zero. You should check the case when ##m=4## by squaring ##(a^2-b^2+abi)## to see what you get.


    Edit: I see that you figured it out as I was posting
     
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