About definition of $a^b$ with $a \in \Bbb{R} \wedge b \in \Bbb{N}$

Question

-- let $a,c \in \mathbb{R}$, and $b \in \Bbb{N}$, with $\Bbb{R}$ is a complete ordered field, $c \triangleq a^b$ if $c=\begin{cases} 1, & \mbox{if } a\neq 0 \wedge b=0\\ 0, & \mbox{if } a=0 \wedge b >0 \\ a \cdot a^{b-1}, & \mbox{if } b>1 \end{cases}$

Is it correct ?

Thanks in advance!!

Answer 1

This leaves $0^0$ undefined. As long as one sticks to integer exponents (and even if not) defining $0^0=1$ is what makes the most sense. Also, $0^b$ with $b>1$ is defined by two of the cases (though it doesn't hurt: the definitions coincide)