# Q. 9 Let ∗ be a binary operation on the set Q of rational numbers as follows:(vi) $a* b = ab^2$Find which of the binary operations are commutative and which are associative.

S seema garhwal

On the set Q ,the operation * is defines as  $a* b = ab^2$ .It is observed that:

For $a,b \in Q$

$1* 2 = 1\times 2^2=1\times 4=4$

$2* 1 = 2\times 1^2=2\times 1=2$

$\therefore$       $1*2\neq 2*1$   for $1,2 \in Q$

Hence, the * operation is not  commutative.

It can be observed that

$(1*2)*3 = (1\times 2^{2})*3 = 4*3 = 4\times 3^{2}=4\times 9=36$

$1*(2*3) = 1*(2\times 3^{2}) = 1*18 = 1\times 18^{2}=1\times 324=324$

$(1*2)*3\neq 1*(2*3)$  for all $1,2,3 \in Q$

The operation * is  not associative.

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