Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.

  2. jcsd
  3. Jun 18, 2008 #2


    User Avatar
    Homework Helper

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


    and n!=n(n-1)!

    so nC1 simplifies to n




    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


    User Avatar
    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook