Question

# Prove the rule of exponents (ab)n=anbn by using principle of mathematical induction for every natural number.

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Solution

## Step (1): Assume given statement Let the given statement be P(n), i. e., P(n):(ab)n=anbn Step (2): Checking statement P(n) for n=1 Put n=1 in P(n), we get P(1):(ab)1=a1b1 ⇒ab=ab Thus P(n) is true for n=1. Step (3): P(n) for n=1 Put n=K in P(n) and assume this is true for some natural number K i.e., P(K):(ab)K=aKbK ⋯(1) Step (4): Checking statement P(n) for n=K+1 Now we shall prove that P(K+1) is true whenever P(K) is true. Now, we have (ab)(K+1) =(ab)K(ab) =(aKbK)(ab) (Using (1)) =(aK⋅a1)(bK⋅b1) =aK+1⋅bK+1 Therefore, P(K+1) is true whenever P(K) is true. Final Answer: Hence, by principle of mathematical induction, P(n) is true for all n∈N.

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