Determine whether the binary operation * on the set N of natural numbers defined by a*b = 2^ab is associative or not.

For the binary operation * on set N to be associative the following equation should hold true

x*(y*z) = (x*y) *z for all x, y, z ∈ N

let us first find x*(y*z)

As a*b = 2^ab

⇒ x*(y*z) = x*(2^yz)

Now let us find (x*y) *z

⇒ (x*y) *z = (2^xy) *z

Comparing (i) and (ii) we have x*(y*z) ≠ (x*y) *z

Hence the binary operation * on set N is not associative