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Binomial theorem for fractional exponents?

  1. Jun 18, 2008 #1
    I am curious, is there any way to use the binomial theorem for fractional exponents? Is there any other way to expand a binomial with a fractional exponent?
    I suppose Newton's theorem is not a way since it requires factorials.

    Thanks!
     
  2. jcsd
  3. Jun 18, 2008 #2

    rock.freak667

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    Homework Helper

    You'd work it out in the same basic way. But just that you'd have an infinite number of terms

    nC1=n!/(n-1)!1!

    and n!=n(n-1)!

    so nC1 simplifies to n

    Similarly

    nC2=n!/(n-2)!2!

    [tex]=\frac{n(n-1)(n-2)!}{(n-2)!2!}=\frac{n(n-1)}{2!}[/tex]

    and so forth for nC3,nC4,etc
     
  4. Jun 19, 2008 #3
    Approximating square roots

    One use, or was so before calculators, is to approximate certain square roots. Take this case,

    [tex]\sqrt{1+a^2} = a +\frac{1}{2a}-\frac{1}{8a^3} +-+[/tex]

    In the case of [tex]\sqrt{101} = 10 + 1/20-1/8000 + -[/tex]

    This is just a little less that 10.05 and can be easily carried out.
     
    Last edited: Jun 19, 2008
  5. Aug 28, 2011 #4
    I also wish to know, is there any way that I can expand the general

    (1+x)^(1/2) = Polynomial_function(x) ??

    (1+x)^(n) easily works out for n>0, n<0,n=0...........but what about for n = 1/m form ??
    m is an Integer.
     
  6. Aug 28, 2011 #5

    uart

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    Science Advisor

    [tex](1+x)^{a}=1+ax+a(a-1)\,\frac{x^{2}}{2!}+a(a-1)(a-2)\,\frac{x^{3}}{3!}+... [/tex]

    This works for integer and non integer "a". So a=1/m is no problems.
     
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