Calculating Fractional Exponent

1. Jul 14, 2010

dimensionless

I have a function f(x) such that

$$f(x) = x^{\frac{a}{b}}$$

where $${\frac{a}{b}}$$ is noninteger. Is there an equation to solve this? A series expansion or something? I've looked around and couldn't find anything.

2. Jul 15, 2010

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 $$x^{a/b} = 7$$, then you could solve for x by taking the b/a power of both sides.

3. Jul 15, 2010

g_edgar

The classical method to do this involves logarithms and exponentials:

$$x^{a/b} = \exp((a/b)\ln(x))$$

4. Jul 17, 2010

dimensionless

To clarify a bit, I'm trying to solve for $$f(x)$$. 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
5. Jul 17, 2010

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
6. Jul 18, 2010

HallsofIvy

Typo: Mark44 meant ex here

7. Jul 18, 2010

Staff: Mentor

Thanks. ex is what I meant. I fixed it in my post.

8. Jul 21, 2010

dimitri151

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?)