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Calculating Fractional Exponent

  1. Jul 14, 2010 #1
    I have a function f(x) such that

    [tex]f(x) = x^{\frac{a}{b}}[/tex]

    where [tex]{\frac{a}{b}}[/tex] is noninteger. Is there an equation to solve this? A series expansion or something? I've looked around and couldn't find anything.
     
  2. jcsd
  3. Jul 15, 2010 #2

    Mark44

    Staff: Mentor

    Solve for what? You don't really have an equation that you can solve for anything. The equation just provides a formula for f(x).

    Now if you had an equation such as [tex]x^{a/b} = 7[/tex], then you could solve for x by taking the b/a power of both sides.
     
  4. Jul 15, 2010 #3
    The classical method to do this involves logarithms and exponentials:

    [tex]x^{a/b} = \exp((a/b)\ln(x))[/tex]
     
  5. Jul 17, 2010 #4
    To clarify a bit, I'm trying to solve for [tex]f(x)[/tex]. By solve, I mean express the solution symbolically and in such a way that the only operations are addition, subtraction, multiplication, and division. In reality, I might consider the use of factorials, sinusoidal functions, special functions, operator functions, etc. to be acceptable. In other words, the question is: how does the calculator solve it? Thanks to g_edgar for answering this question.
     
    Last edited: Jul 17, 2010
  6. Jul 17, 2010 #5

    Mark44

    Staff: Mentor

    That's not "solving" for f(x). As I already said, the equation for f(x) is merely a definition of its formula. What you want to do is write the formula in a different form.
    Edit: Fixed typo: e2 --> ex
    What you're asking about is answered in the part of calculus that deals with power series, such as Taylor and Maclaurin series, and Fourier series, to name a few. A function such as ex has a Maclaurin series 1 + x + x2/2! + x3/3! + ... + xn/n! + ... As you can see, the series representation consists only of addition and multiplication (plus factorials).

    As I understand things, calculators use a technique similar to this but not exactly the same, combined with lookup tables, to calculate the various functions that are on a scientific calculator. It's been a long time since I thought about it, but the acronym CORDIC fits in here somehow.
     
    Last edited: Jul 18, 2010
  7. Jul 18, 2010 #6

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Typo: Mark44 meant ex here

     
  8. Jul 18, 2010 #7

    Mark44

    Staff: Mentor

    Thanks. ex is what I meant. I fixed it in my post.
     
  9. Jul 21, 2010 #8
    xa/b=(1+x-1)a/b=1+(a/b)(x-1)+(a/b)(a/b-1)/2!(x-1)2
    +(a/b)(a/b-1)(a/b-2)/3!(x-1)3...
    (Sorry, cant do better latex..any good guides?)
     
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