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Rational Exponents

A rational number is a real number that can be expressed as a ratio of two finite integers:

$\displaystyle x = \frac{L}{M}, \quad L\in{\bf Z},\quad M\in{\bf Z}
$

Applying property (2) of exponents, we have

$\displaystyle a^x = a^{L/M} = \left(a^{\frac{1}{M}}\right)^L.
$

Thus, the only thing new is $ a^{1/M}$ . Since

$\displaystyle \left(a^{\frac{1}{M}}\right)^M = a^{\frac{M}{M}} = a
$

we see that $ a^{1/M}$ is the $ M$ th root of $ a$ . This is sometimes written

$\displaystyle \zbox {a^{\frac{1}{M}} \isdef \sqrt[M]{a}.}
$

The $ M$ th root of a real (or complex) number is not unique. As we all know, square roots give two values (e.g., $ \sqrt{4}=\pm2$ ). In the general case of $ M$ th roots, there are $ M$ distinct values, in general. After proving Euler's identity, it will be easy to find them all (see §3.11). As an example, $ \sqrt[4]{1}=1$ , $ -1$ , $ j$ , and $ -j$ , since $ 1^4=(-1)^4=j^4=(-j)^4=1$ .


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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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