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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
. Since
we see that
is the
th root of
.
This is sometimes written
The
th root of a real (or complex) number is not unique. As we all
know, square roots give two values (e.g.,
). In the
general case of
th roots, there are
distinct values, in
general. After proving Euler's identity, it will be easy to find them
all (see §3.11). As an example,
,
,
,
and
, since
.
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