Power of a product rule for exponents

Students will learn what the power of a product rule for exponents is. Then students will see why it's true, by expanding powers of products, then simplifying, as seen here. Then students will learn how to use the power of a product rule. Problems may include negative powers. Here's an interesting problem: Prove the product of two square numbers is square. After that students will be warned that \((x + y)^n \ne x^n + y^n.\) This mistake is called the freshman's dream. To prevent this mistake, students will see two disproofs. The first proof is by counterexample. It's enough to show \((1 + 1)^2 \ne 1^2 + 1^2.\) The second proof is done by algebra tiles, or equivalently, by algebra. By doing or seeing the second proof, students will also learn the conditions under which \((x + y)^n = x^n + y^n.\)