Q.Prove the rule of exponents $(ab)^n = a^n b^n$ by using principle of mathematical induction for every natural number.

Let $P(n)$ be the statement $(ab)^n=a^nb^n$, for real numbers $a,b$.

Base case: For $n=1$,

So $P(1)$ is true.

Inductive step: Assume $P(k)$ is true for some $k\ge1$:

We must show $(ab)^{k+1}=a^{k+1}b^{k+1}$.

By the definition of exponentiation,

Substitute the induction hypothesis:

Using commutativity and associativity of multiplication to regroup the factors: …

Foundational algebraic identity proved by induction directly from the recursive definition $(ab)^{k+1}=(ab)^k(ab)$, substituting the hypothesis and regro …

• Assuming the result to prove commutativity of multiplication as if it were self-evident from the statement, rather than explicitly invoking it as the justification for regrouping $a^k\cdot a\cdot b^k\cdot b$ into $a^{k+1}b^{k+1}$. …