00A05-Introducing0thPower.tex

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\pmtitle{introducing 0th power}
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Let $a$ 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
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a^n \cdot 1=a^n=a^{n+0}=a^n\cdot a^0,
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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.
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