Rational Exponents

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

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

Applying property (2) of exponents, we have

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

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

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

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

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

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 $1^4=(-1)^4=j^4=(-j)^4=1$.