Answer :

To prove: * is neither commutative nor associative

Let us assume that * is commutative

⇒ a^{b} = b^{a} for all a,b N

This is valid only for a = b

For example take a = 1, b = 2

1^{2} = 1 and 2^{1} = 2

So * is not commutative

Let us assume that * is associative

⇒ (a^{b})^{c} for all a,b,c N

for all a,b,c N

This is valid in the following cases:

(i) a = 1

(ii) b = 0

(iii) bc = b^{c}

For example, let a = 2,b = 1,c = 3

a^{bc} = 2^{(1 × 3)} = 2^{3} = 8

= 2

So * is not associative

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