Prove the following by the principle of mathematical induction:`\ (a b)^n=a^n b^n` for all `n in Ndot`

Explanation

Let `P(n)` be the given statement


i.e., `P(n):(a b)^{n}=a^{n} b^{n}`


We note that `P(n)` is true for `n=1` since `(a b)^{1}=a^{1} b^{1}`.


Let `P(k)` be true, i.e.,



`(a b)^{k}=a^{k} b^{k} ldots(1)`



We shall now prove that `P(k+1)` is true whenever `P(k)` is true.


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