Use mathematical induction to prove that each statement is true for every positive integer n. (ab)ⁿ = aⁿ bⁿ

Verified step by step guidance

1. Step 1: **State the proposition** you want to prove using mathematical induction. Let \( P(n) \) be the statement \( (ab)^n = a^n b^n \) for a positive integer \( n \).
2. Step 2: **Base case:** Verify that \( P(1) \) is true. Substitute \( n = 1 \) into the statement to check if \( (ab)^1 = a^1 b^1 \) holds.
3. Step 3: **Inductive hypothesis:** Assume that \( P(k) \) is true for some positive integer \( k \), i.e., assume \( (ab)^k = a^k b^k \).
4. Step 4: **Inductive step:** Use the inductive hypothesis to prove that \( P(k+1) \) is true. Start with \( (ab)^{k+1} = (ab)^k (ab) \) and then substitute the inductive hypothesis to rewrite it as \( a^k b^k (ab) \).
5. Step 5: **Simplify the expression** from Step 4 by using the associative and commutative properties of multiplication to show that \( a^k b^k (ab) = a^{k+1} b^{k+1} \), thus proving \( P(k+1) \). Conclude that by mathematical induction, the statement is true for all positive integers \( n \).

Key Concepts

Mathematical Induction

Mathematical induction is a proof technique used to establish that a statement holds for all positive integers. It involves two steps: proving the base case (usually n=1) is true, and then proving that if the statement holds for an arbitrary integer k, it also holds for k+1. This creates a chain of truth for all n.

Properties of Exponents

The properties of exponents describe how to manipulate expressions involving powers. One key property is that (ab)^n = a^n * b^n, meaning the nth power of a product equals the product of the nth powers. Understanding this property is essential to prove the given statement using induction.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using algebraic rules. In this proof, it is necessary to rewrite (ab)^(k+1) as (ab)^k * (ab) and then apply the induction hypothesis and exponent rules to show equality. Mastery of these skills helps in constructing clear and logical proofs.