Students will learn what the product of powers rule for exponents is, and how to use it. Problems may include negative powers. To understand why the product of powers rule works, students will start by reviewing the distributive property of multiplication over addition. That is, the distributive property works because multiplication is repeated addition. Consider that \(2 \cdot 3 + 2 \cdot 4\) is the sum of 3 2's and 4 2's, which is the same as 7 2's. Mathematically, we say \(2 \cdot 3 + 2\cdot 4 = 2 \cdot 7.\) More generally, but using the same reasoning, we find \(x \cdot y + x \cdot z = x(y + z).\) In the same way, \(2^3 \cdot 2^4 = 2^7,\) because exponentiation is repeated multiplication. Generalizing this, we get \(x^y \cdot x^z = x^{y + z}.\)