introducing 0th powerIntroducing0thPower2003-01-31 04:04:272004-11-08 17:21:23Topic% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLet $a$ be a number not equal to zero. Then for all $n \in \mathbb{N}$, we have that $a^n$ is the product of $a$ with itself $n$ \PMlinkescapetext{times}. Using the fact that the integer 1 is a multiplicative identity, ($a\cdot 1=a$ for any $a$), we can write
\begin{displaymath}
a^n \cdot 1=a^n=a^{n+0}=a^n\cdot a^0,
\end{displaymath}
where we have used the properties of exponents under multiplication. Now, after canceling a factor of $a^n$ from both sides of the above equation, we derive that $a^0=1$ for any non-zero number.