## 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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