What is the definition of raising a number to the zeroth power ($x^0$)? I know that many people say that "anything raised to the zeroth power is one" but this is clearly not true since $0^0$ is $undefined$. How then do mathematicians define $x^0$ such that for all real numbers not equal to $0$, $x^0=1$?
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I think it may have to do with the fact that 1 is the multiplicative identity of a field. I'm commenting for a notification . – Yunus Syed Oct 12 '15 at 03:43
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We mathematicians bend the rules and change them anyway we want:) Ok, essentially, we accept $x^0=1$ for all $x$ not zero There are different levels of proof for this. – imranfat Oct 12 '15 at 03:43
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1Actually, many mathematicians define $0^0=1$. Ter are strong reasons to do this - any empty product is $1$. The product of $0$ 1's is the same as the product of $0$ 0's. – Thomas Andrews Oct 12 '15 at 03:47
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You may find this interesting:http://www.askamathematician.com/2010/12/q-what-does-00-zero-raised-to-the-zeroth-power-equal-why-do-mathematicians-and-high-school-teachers-disagree/ – NoChance Oct 12 '15 at 03:58
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Depends how strong of a proof you are looking for, but one way you could think of it is $$x^0 = x^{1-1} = \frac{x^1}{x^1} = \frac{x}{x} = 1$$ It may not intuitively be equal to $1$, but it is necessarily equal to $1$.
graydad
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