wikipedia says for positive integer exponents its defined by the initial condition
$b^1=b$ and the recurrence relation $b^{n+1} = b^n*b$ and for negative integers its defined as $a^{-n}=\frac{1}{a^n}$.
so is this definition of $a^x$ for all integers, as you are clearly describing the nature of this notation for all integer $x$, but $0$?
also is $a^{-n}=\frac{1}{a^n}$ like an axiom as its just taken to be true or can you prove it. Thanks (hopefully this is clearer than my earlier question)
Simply $$b^0=1$$ $$b^{n+1}=b(b^n)$$ $$b^{-n}=\frac{1}{b^n}$$ In terms of proving your last statement, consider that $\frac{b^n}{b^n}=1=b^0$ and $b^xb^y=b^{x+y}$. Then $x+y=0 \iff y=-x$ and you can establish that $b^{-n}=\frac{1}{b^n}$