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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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