Welcome to Rules of Integer Exponents, the second course in your learning path! In the previous course, you built a solid understanding of what exponents mean, how negative bases and signs behave, and what happens with zero and negative exponents. Now it's time to put that foundation to work and learn the shortcuts that make exponent arithmetic fast and efficient. In this first lesson, we focus on the Product Rule — the rule that tells you exactly what happens when you multiply two powers that share the same base.

Before we introduce anything new, let's ground ourselves in one key idea. An exponent tells you how many times to use the base as a factor. For example, $2^3$ means $2 \times 2 \times 2$ (three factors of 2), and $5^4$ means (four factors of 5).

Let's see what happens when we multiply two powers that have the same base. Consider $2^3 \times 2^2$. Expanding each power into its factors gives us:

Here is the general rule we just discovered:

When multiplying two powers that share the same base, keep the base and add the exponents. A couple of quick examples:

The product rule isn't limited to positive exponents. Recall that $a^0 = 1$ for any nonzero base, and a negative exponent gives a reciprocal (for instance, $2^{-3} = \frac{1}{2^3}$). The great news is that the rule works exactly the same way — just add the exponents, even when they are zero or negative.

The product rule shows up naturally whenever same-base quantities multiply together. Let's walk through three real-world scenarios.

Biology — bacterial growth. Suppose a biology student counts $3^2$ bacteria on a slide. After incubation, the colony grows by a factor of $3^4$. The total count is:

Let's recap the big idea. When you multiply two or more powers that share the same base, you keep the base and add the exponents: $a^m \times a^n = a^{m+n}$, and more generally $$. This shortcut follows directly from viewing exponents as a count of repeated factors, and it works whether the exponents are positive, zero, or negative.