Question

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

Answer 1

Let P(n) be the given statement

i.e., P(n):(ab)ⁿ=aⁿbⁿ

We note that P(n) is true for n=1 since (ab)¹=a¹b¹.

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

(ab)^k=a^kb^k----------(1)

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

Now, we have

=(ab)^k+1=(ab)^k(ab)

=(a^kb^k)(ab) [ by (1) ]

=(a^k.a¹)(b^k.b¹)=a^k+1.b^k+1

Therefore, P(k+1) is also true whenever P(k) is true. 

Hence by principle of mathematical induction, P(n) is true for all n∈N.