Question

16. By using principle of mathematical induction for every natural number.Prove the rule
of exponents (ab)^n = a^n b^n​

Answer 1

We have to prove the rule of exponent (ab)ⁿ = aⁿ bⁿ by using principle of mathematical induction for every natural number.

proof : let P(n) : (ab)ⁿ = aⁿ bⁿ

so p(n) is true for n = 1 since p(1) = (ab)¹ = a¹ b¹ = ab.....(1) which is true.

let pp(k) : (ab)^k = a^k b^k.....(2) , is true for some positive integer k.

now we shall now prove that p(k + 1) is true whenever p(k) is true.

p(k + 1) : LHS = (ab)^(k + 1) = (ab)^k × ab¹

= a^k b^k × a¹ b¹ [ from equations (1) and (2) ]

= a^k a¹ × b^k b¹

= a^(k + 1) b^(k + 1)

Therefore p(k + 1) : (ab)^(k + 1) = a^(k + 1) b^(k + 1) which is true whenever p(k) is true.

Therefore (ab)ⁿ = aⁿ bⁿ is true for every natural number.