Rational Exponents
A
rational
number is a real number that can be expressed as
a ratio of two finite integers:
Applying property (2) of exponents, we have
Thus, the only thing new is
![$ a^{1/M}$](http://www.dsprelated.com/josimages_new/mdft/img248.png)
. Since
we see that
![$ a^{1/M}$](http://www.dsprelated.com/josimages_new/mdft/img248.png)
is the
![$ M$](http://www.dsprelated.com/josimages_new/mdft/img244.png)
th root of
![$ a$](http://www.dsprelated.com/josimages_new/mdft/img236.png)
.
This is sometimes written
The
![$ M$](http://www.dsprelated.com/josimages_new/mdft/img244.png)
th root of a real (or complex) number is not unique. As we all
know, square roots give two values (
e.g.,
![$ \sqrt{4}=\pm2$](http://www.dsprelated.com/josimages_new/mdft/img251.png)
). In the
general case of
![$ M$](http://www.dsprelated.com/josimages_new/mdft/img244.png)
th roots, there are
![$ M$](http://www.dsprelated.com/josimages_new/mdft/img244.png)
distinct values, in
general. After proving
Euler's identity, it will be easy to find them
all (see §
3.11). As an example,
![$ \sqrt[4]{1}=1$](http://www.dsprelated.com/josimages_new/mdft/img252.png)
,
![$ -1$](http://www.dsprelated.com/josimages_new/mdft/img160.png)
,
![$ j$](http://www.dsprelated.com/josimages_new/mdft/img89.png)
,
and
![$ -j$](http://www.dsprelated.com/josimages_new/mdft/img203.png)
, since
![$ 1^4=(-1)^4=j^4=(-j)^4=1$](http://www.dsprelated.com/josimages_new/mdft/img253.png)
.
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