Rational Exponents

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

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

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