Question

Let * be a binary operation on the set $Q$ of rational numbers as follows:
$a\ast b={ab}^2$
Find which of the binary operations are commutative and which are associative.

Answer 1

$a\ast b=a{b}^2$ is a binary operation on $Q$.

(1) Associative : (a\ast b)\ast c=ab^{2}+c=ab^{2}+c^{2}$

$a\ast (b\ast c)=a+\left(b{c}^2\right)\ne (a\ast b)\ast c$

Not Associative.

(2) Commative : $a\ast b=a{b}^2$

$b\ast a=b{a}^2\ne a\ast b$

Not Commutative.