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Tedious Simplification: Show expression is real

  1. Mar 29, 2012 #1
    After doing A LOT of simplification on a complicated expression I am now at a point where my own skills can't take me any further.

    My problem is that I have an expression, which I am almost certain must be real, but it contains the imaginary units here and there. I have tried using Maple and Mathematica with different assumptions, but it does not simplify very much.

    Now, I do not expect you to do my tedious algebra. However, I was hoping that one of you might know of a technique (in Maple/Mathematica/Hand) that could help me?

    My expression is this:

    [itex]{\frac {\sqrt [3]{2}n \left( -1+n \right) \left( -1+2\,n \right)
    \left( 2\,\sqrt [3]{-2}+4\,\sqrt [3]{-2} \left( -1+n \right) n+
    \left( -1-2\,n \left( 1+ \left( -3+n \right) n \right) +i\sqrt {1-16
    \,n+44\,{n}^{2}-44\,{n}^{3}+28\,{n}^{4}-24\,{n}^{5}+12\,{n}^{6}}
    \right) ^{2/3}-i\sqrt {3} \left( -1-2\,n \left( 1+ \left( -3+n
    \right) n \right) +i\sqrt {1-16\,n+44\,{n}^{2}-44\,{n}^{3}+28\,{n}^{4
    }-24\,{n}^{5}+12\,{n}^{6}} \right) ^{2/3} \right) }{\sqrt [3]{-1-2\,n
    \left( 1+ \left( -3+n \right) n \right) +i\sqrt {1-16\,n+44\,{n}^{2}-
    44\,{n}^{3}+28\,{n}^{4}-24\,{n}^{5}+12\,{n}^{6}}} \left( -8+16\,n
    \right) }} [/itex]
    where n is an interger larger than 1.

    as you can see the same terms appear many places.


    Any help will be greatly appreciated.

    Maple format:
    Code (Text):
    2^(1/3)*n*(-1+n)*(-1+2*n)*(2*(-2)^(1/3)+4*(-2)^(1/3)*(-1+n)*n+(-1-2*n*(1+(-3+n)*n)+I*sqrt(1-16*n+44*n^2-44*n^3+28*n^4-24*n^5+12*n^6))^(2/3)-I*sqrt(3)*(-1-2*n*(1+(-3+n)*n)+I*sqrt(1-16*n+44*n^2-44*n^3+28*n^4-24*n^5+12*n^6))^(2/3))/((-1-2*n*(1+(-3+n)*n)+I*sqrt(1-16*n+44*n^2-44*n^3+28*n^4-24*n^5+12*n^6))^(1/3)*(-8+16*n))
    Mathematica format:
    Code (Text):
    (2^(1/3) (-1 + n) n (-1 + 2 n) (2 (-2)^(1/3) +
         4 (-2)^(1/3) (-1 + n) n + (-1 - 2 n (1 + (-3 + n) n) +
           I Sqrt[1 - 16 n + 44 n^2 - 44 n^3 + 28 n^4 - 24 n^5 +
             12 n^6])^(2/3) -
         I Sqrt[3] (-1 - 2 n (1 + (-3 + n) n) +
            I Sqrt[1 - 16 n + 44 n^2 - 44 n^3 + 28 n^4 - 24 n^5 +
              12 n^6])^(2/3)))/((-8 + 16 n) (-1 - 2 n (1 + (-3 + n) n) +
         I Sqrt[1 - 16 n + 44 n^2 - 44 n^3 + 28 n^4 - 24 n^5 + 12 n^6])^(
       1/3))
    Thanks in advance
     
    Last edited: Mar 29, 2012
  2. jcsd
  3. Mar 29, 2012 #2
    Here's one idea:

    If the number is pure real, then you can multiply the i's by -1 and after this, you should have the same number as before.
     
  4. Mar 29, 2012 #3
    Thanks. Yes, that is a good idea. However, the expression is so complicated that it doesn't really help. I can't show the expressions are the same.

    And actually, which I should have clarified, I do need the simplified expression.
     
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