Since I learned exponentiation in School, me and my classmates had this one question stuck in our brains. Why is $a^0=1$ and not $0$. Wouldn't it make more sense if it equaled to $0$, since $a$ is multiplied by itself $0$ times, it should equal $0$. Can somebody please explain to me how and why this works?
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2The choice $a^0=1$ is the only one which guarantees that $a^0a^x=a^{0+x}$ for every $x$ (as has been explained multiple times on the site). – Did Jan 22 '17 at 12:05
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2Let $a \neq 0 ,m \in R $ now $\frac{a^m}{a^m}=1$ so $$\frac{a^m}{a^m}=a^{m-m}=a^0=1$$ – Khosrotash Jan 22 '17 at 12:09