The laws of exponents are a set of five rules that show us how to perform some basic operations using exponents.

Mathematically they are defined as follows:

Let a and b be real numbers and m and n be positive integers. Then the following laws hold:

Notice that we have taken our exponents to be positive integers. This is the most intuitive choice because we know that,

where n is a positive integer. We can also use the first law of exponents to define the zero and negative exponents,

In (1), a ⁰ behaves like the number 1, and we define a ⁰ = 1. In (2), a ⁻ⁿ behaves like the reciprocal of a ⁿ and we write,

The laws of exponents provide a set of rules that can be used to simplify complex expressions that contain exponents.

These rules are important when simplifying expressions involving exponents. For example, we can use laws of exponents to make the following simplification,

The purpose of simplification will become evident as you begin to solve problems involving exponents; sometimes these simplifications are necessary to see the next step in solving a problem.

A common mistake you should avoid

It is important to recognize when no simplification can be made. Notice that the laws of exponents do not involve sums or differences of terms with exponents.

You might be tempted to write,

However, this statement is incorrect because the sum of two exponents cannot be further simplified. Many of the applications we discuss will require that you simplify an expression that contains exponents. When doing so, you must keep in mind the rules we have described in this section.

*****

In the next section we will begin an exploration of the graphs of exponential functions.