The Exponent Zero

How should we define $a^0$ in a manner consistent with the properties of integer exponents? Multiplying it by $a$ gives

$$a^0 \cdot a = a^0 a^1 = a^{0+1} = a^1 = a$$

by property (1) of exponents. Solving $a^0 \cdot a = a$ for $a^0$ then gives

$$\zbox {a^0 = 1.}$$