$a*b = \sqrt{ab} \space a,b \in \mathbb{R}$
It is easy enough to show it is communitive and not associative:
$a*b=\sqrt{ab}=\sqrt{ba}=b*a$ thus commutative
$(a*b)*c= \sqrt{ab\sqrt{c}} \neq\sqrt{a\sqrt{bc}}= a*(b*c)$ not associative
to find an identity element I need to find an element $e$ S.T. $a*e =a$ Well if we let $b=a$ then a*b = $\sqrt{a^2}=a$ That follows right?
As for an inverse I need something that gives me back...well im not sure
You're close!
One thing to note is that you must provide a counterexample to claim $a * b$ is not asociative. That is, can you provide three actual numbers $a,b,c \in \mathbb{R}$ so that $a*(b*c) \neq (a*b)*c$?
As for the identity, $e$ is not allowed to vary! It must be ONE number so that $a*e = e*a = a$. Your argument actually shows that no such $e$ exists, since for $a \neq e$, we must have $a*e \neq a$.
Finally, since there is no identity, there is no hope of inverses. This is another reason to want one $e$ which works as an identity for every $a$, if you don't have it, then it's not clear what $a * a^{-1}$ should be.
I hope this helps ^_^