In Exercises 25–34, 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: Understand the statement to prove: For every positive integer n, (ab)^n = a^n b^n. This means that raising the product of a and b to the power n is the same as raising each factor to the power n and then multiplying.
2. Step 2: Base Case: Verify the statement for n = 1. Substitute n = 1 into the equation to check if (ab)^1 = a^1 b^1 holds true.
3. Step 3: Inductive Hypothesis: Assume the statement is true for some positive integer k, that is, assume (ab)^k = a^k b^k.
4. Step 4: Inductive Step: Use the inductive hypothesis to prove the statement for n = k + 1. Express (ab)^{k+1} as (ab)^k * (ab) and then apply the inductive hypothesis to rewrite (ab)^k as a^k b^k.
5. Step 5: Simplify the expression (ab)^{k+1} = a^k b^k * ab = a^k * a * b^k * b = a^{k+1} b^{k+1}, which completes the inductive step and proves the statement 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

Properties of exponents describe how powers behave under multiplication and other operations. For example, (ab)^n = a^n b^n means raising a product to a power is the same as raising each factor to that power separately. Understanding these rules is essential to manipulate and simplify expressions involving exponents.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using algebraic rules. In this proof, it is important to correctly apply multiplication and exponent rules to show the inductive step holds. Careful manipulation ensures the logical flow from the assumption to the conclusion.