Exponentiation

$a^n$ is a shorthand for repeated multiplication. For example, $a^3 = a \times a \times a$ . In other words, to compute the value of $a^3$ we multiply three $a$ ‘s. $a^3 \times a^2 = (a \times a \times a) \times (a \times a) = a^5 = a^{3+2}$ . We can generalize: $a^{m+n} = a^m \times a^n$ . Since $\frac{a^5}{a^3} = \frac{a \times a \times a \times a \times a}{a \times a \times a} = a \times a = a^2 = a^{5-3}$ , we can generalize: $a^{m-n} = \frac{a^m}{a^n}$ . $a^0 = a^{n-n} = \frac{a^n}{a^n} = 1$ . However, $0^0 = 0^{n-n} = \frac{0^n}{0^n} = \frac{0}{0}$ . Since $\frac{0}{0}$ is undefined, $0^0$ is also undefined. $a^{-3} = a^{0-3} = \frac{a^0}{a^3} = \frac{1}{a^3}$ . Thus $a^{-3}$ is the reciprocal of $a^3$ . Obviously, $0^{-3}$ is undefined.

What about $a^{\frac{1}{3}}$ ? We observe that, $a^{\frac{1}{3}} \times a^{\frac{1}{3}} \times a^{\frac{1}{3}} = a^{\frac{1}{3} + \frac{1}{3} + \frac{1}{3}} = a^{3 \times \frac{1}{3}} = a^1 = a$ . It follows that the product of three $a^{\frac{1}{3}}$ ‘s is $a$ . In other words, $a^{\frac{1}{3}}$ is the cube root of $a$ . We can generalize: $a^{\frac{1}{n}} = \sqrt[n]{a}$ .