Does $a ∗ b = a ^b$ define a binary operation on the set $R^∗$ of nonzero real numbers?

Question

Does $a ∗ b = a^b$ define a binary operation on the set $R^∗$ of nonzero real numbers?

It seems to me that it is defined since it cannot be $0$.

Answer 1

No, it doesn't. A binary operation must be closed, i.e. if $a,b\in \mathbb{R}\setminus\{0\}$, then $a*b\in \mathbb{R}\setminus\{0\}$. But if we take $a=-1$ and $b=\frac{1}{2}$, we have $\sqrt{-1}$ which is not in $\mathbb{R}\setminus\{0\}$.