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Radical fraction expression

  1. Apr 17, 2004 #1
    Simplify (y^-3 - x^-3) / (xy^-1 + x^-1y + 1)

    better picture of it here

    I tried flipping the variables with negative exponents to the numerator and denominator, but then had no idea what to do next :frown:
    Last edited by a moderator: Apr 20, 2017
  2. jcsd
  3. Apr 17, 2004 #2
    I doubt you will get better simplification than that, you could multiply top and bottom by xy to get a nice denominator.
  4. Apr 17, 2004 #3
    It certainly does get simpler!

    First consider the numerator:

    y^-3 - x^-3 = 1/y^3 - 1/x^3 = (x^3 - y^3)/(x^3 y^3) = (x - y)(x^2 + y^2 + xy)/(x^3 y^3)

    Now the denominator:

    x y^-1 + y x^-1 + 1 = x/y + y/x + 1 = (x^2 + y^2 + xy)/(xy)

    Combine them:

    (x - y)(x^2 + y^2 + xy)/(x^3 y^3)(xy)/(x^2 + y^2 + xy)

    The x^2 + y^2 + xy terms on top and bottom cancel, and one of each of the powers of x and y on the bottom cancel to yield a final simplification of:

    [tex]\frac{x - y}{x^2y^2}[/tex]

    Much nicer!

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