Given $a \ast b = ab^2$

let us define e as the identity element of the operation *

By definition of identity element a*e =e*a = a

But from definition of * operation $a \ast e = ae^2$ =>$ a=ae^2$, and which imples that $e^2 =1$

Also ,$e \ast a = ea^2$ =>$ a=e a^2$ implies that e=1/a

from the above , we get two values for e

This is not possible as the identity is unique. Therefore *operation defined by $a \ast b = ab^2$ has no identity.