Given that * is a binary operation on N defined by a*b = a^b for all a,b∈N.

We know that commutative property is p*q = q*p, where * is a binary operation.

Let’s check the commutativity of given binary operation:

⇒ a*b = a^b

⇒ b*a = b^a

⇒ b*a≠a*b

∴ Commutative property doen’t holds for given binary operation ‘*’ on ‘N’.

We know that associative property is (p*q)*r = p*(q*r)

Let’s check the associativity of given binary operation:

⇒ (a*b)*c = (a^b)*c

⇒ (a*b)*c = (a^b)^c

⇒ (a*b)*c = a^bc ...... (1)

⇒ a*(b*c) = a*(b^c)

⇒ ...... (2)

From (1) and (2) we can clearly say that associativity doesn’t hold for the binary operation ‘*’ on ‘N’.