Power with Rational Exponent
1. Motivation: Why Rational Exponents?
You already know how to work with integer exponents: $a^2$, $a^{-3}$, $a^0 = 1$, etc. But what does $a^{1/2}$ or $a^{2/3}$ mean?
Key Idea: When we introduce fractional (rational) exponents, we require that all properties of integer exponents still hold. In particular, the power-of-a-power rule must work:
$$(a^m)^n = a^{m \cdot n}$$
Let's see what this forces. Suppose we want $(a^{1/2})^2 = a^{(1/2) \cdot 2} = a^1 = a$. This means $a^{1/2}$ must be a number whose square equals $a$. That's exactly $\sqrt{a}$!
Similarly, $(a^{1/n})^n = a^1 = a$, so $a^{1/n} = \sqrt[n]{a}$.
⚠️ Important Restriction: The base must be positive ($a > 0$) for rational exponents to be well-defined. We'll see why shortly.
2. Definition of $a^{m/n}$
Definition
For $a > 0$ and a rational number $r = \dfrac{m}{n}$, where $m$ is an integer and $n$ is a natural number:
$$\boxed{a^r = a^{m/n} = \sqrt[n]{a^m}}$$Additionally, if $m$ and $n$ are natural numbers, then $0^{m/n} = 0$.
Examples:
| Expression | Conversion | Value |
|---|---|---|
| $5^{2/5}$ | $\sqrt[5]{5^2} = \sqrt[5]{25}$ | $\sqrt[5]{25} \approx 1.904$ |
| $2^{0.5} = 2^{1/2}$ | $\sqrt{2}$ | $\approx 1.414$ |
| $10^{-1} = 10^{-1/1}$ | $\sqrt[1]{10^{-1}} = \dfrac{1}{10}$ | $0.1$ |
| $8^{2/3}$ | $\sqrt[3]{8^2} = \sqrt[3]{64}$ | $4$ |
| $27^{-1/3}$ | $\dfrac{1}{\sqrt[3]{27}} = \dfrac{1}{3}$ | $\approx 0.333$ |