May 25, 2016

This month I want to focus on a subject many students find challenging: exponents. This post is a basic overview of exponent properties and is a good place to start when preparing for this topic.

Three Rules

When working with exponents, there are three major rules that govern the simplification of expressions:

The first rule says that if you are multiplying two expressions with the same base, then you can keep that base and add the exponents. The second rule is similar to the first, but this time, if you divide expressions with the same base, you can subtract the exponents. And lastly, if you have an exponent raised to another exponent, you can multiply the exponents. These rules hold firm for any exponents (whole numbers, negative numbers, fractions, zero). Notice there are no rules for adding and subtracting these types of expressions. The only time you can add or subtract is when you have the exact same expression. An example: . Notice that  has the same base and exponent, which allows us to add.

Applying the Rules

Simplify the following expression:

Step 1: Identify any of the three exponent rules.

Our example has division, but the addition in the numerator does not have an exponent rule. We also can’t add the two terms together since they are not exactly the same (same base, different exponent). We must rearrange the statement to apply the division rule.

Step 2: Separate terms if necessary.

We can change the fraction to look like:

Now that the terms are separated, we can apply the division exponent rule.

Step 3: Apply exponent rules.

This cannot be further simplified since the terms are not exactly the same.

The Effect of Negative Exponents

Negative exponents are often not fully understood. The main concept to remember is negative exponents do not create negative answers. Let’s look at how negative exponents are defined:

Notice the pattern: When you subtract 1 from the exponent, the answer is divided by 3. This pattern continues as you keep subtracting 1 from the exponent.

This is why a number to the zero power is always 1, and it shows the connection negative exponents have with division.

When working with negative exponents, you want to mentally connect them with fractions.

Three basic examples:

I often tell students that “flipping the fraction” gets rid of the negative sign on the exponent.

Fractional Exponents

The last big topic in exponents is fractional exponents. Similar to how negative exponents should have an immediate connection to fractions, fractional exponents should have an immediate connection to radicals. All fraction exponents are radicals in disguise.

This is equivalent to

Let’s simplify another example:

Step 1: The negative exponent tells us to “flip the fraction”:

Step 2: Change the fraction into its radical form: