Determine whether or not each of the definitions of `**` given below gives a binary operation. In the event that `**` is not a binary operation, give justification for this
on `R`, define `**` by `a**b=ab^2`

Explanation

`a**b=ab^2, b**a=ba^2`
`a**b ne b**a AA a, b in Q`
`therefore **` is not commutative
`a**(b**c)=a**bc^2`
`=a(bc^2)^2=ab^2c^4`
`(a**b)**c=(ab^2)**c=ab^2c^2`
`therefore a**(b**c)ne(a**b)**c AA a,b,c in Q`
`therefore **` is not associative