The Exponent Zero

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

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

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

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


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