Why does any non-zero expression raised to the power of 0 equal 1?

Question

How would you answer? When I opened Photomath, it said any non-zero expression raised to the power of zero equals one.

Answer 1

Basically it's a definition. It's not a deductive consequence of simpler facts; it's how we choose to interpret $a^0$.

This is not to say that it is just random, though. It's a particularly useful definition because it's the unique value for $a^0$ that makes makes the rule $$ a^{b+1} = a\cdot a^b $$ hold when we set $b=0$. Since that rule definitely holds in the "obvious" case when $a$ and $b$ are positive integers, it is useful to get it to hold as widely as we can make it. This is also the motivation for later defining $a^{-n} = \dfrac1{a^n}$.

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This reasoning does not hold with the same force in the case where $a=0$, because then the above rule just requires $0^1 = 0\cdot 0^0$, which will be true automatically no matter what we take $0^0$ to be. That's the reason why the definition in some books deliberately only defines $a^0$ for nonzero $a$. It is actually very common (but not universal) to define the value of $0^0$ to be $1$ too, but that is for more subtle reasons than this.