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.