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

Note: This is part 6 of a 6 part question, split as 6 separate questions here.

Toolbox:
• An operation $\ast$ on $A$ is commutative if $a\ast b = b \ast a\; \forall \; a, b \in A$
• An operation $\ast$ on $A$ is associative if $a\ast ( b \ast c) = (a \ast b) \ast c\; \forall \; a, b, c \in A$
Given a binary operation $\ast$ in Q defined by $a \ast b=ab^2$:
$\textbf {Step 1: Checking if the operation is Commutative}$:
For an operation $\ast$ to be commutative $a\ast b = b \ast a$.
$a \ast b = ab^2$, but $b \ast a = ba^2$, which are not equal unless $a=b=1$. Hence the operation $\ast$ is not commutative.
$\textbf {Step 2: Checking if the operation is Associative}$:
For an operation $\ast$ on $A$ is associative $a\ast ( b \ast c) = (a \ast b) \ast c\; \forall \; a, b, c \in A$
$a \ast (b \ast c) = a \ast (bc^2) = a(bc^2)^2 = ab^2c^4$ and $(a \ast b) \ast c = (ab^2) \ast c = ab^2c^2$
Since $a\ast ( b \ast c) \neq (a \ast b) \ast c$, the operation $\ast$ is not associative.
edited Mar 19, 2013