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# introducing 0th power

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 $n$ times. Using the fact that the integer 1 is a multiplicative identity^{}, ($a\cdot 1=a$ for any $a$), we can write

$a^{n}\cdot 1=a^{n}=a^{{n+0}}=a^{n}\cdot a^{0},$ |

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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