Simplifying Powers of Fractions

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

This lesson will go over what powers of fractions are and some basic vocabulary. We will then see two different ways of simplifying powers of fractions and look at some examples showing each method.

Powers of Fractions

Have you ever been making a meal or snack by following a recipe, but you don't want to make as many servings as the recipe makes? Suppose you are making dinner for you and two of your friends. You are following a recipe that makes four servings, but you only want three servings, so you want to make 3/4 of the recipe. To do this, you just need to only use 3/4 of the amount of each of the ingredients called for. You notice that the recipe calls for 3/4 cup of chopped onions, so you need to add 3/4 of 3/4 of a cup of chopped onion. In other words, you need to add (3/4) * (3/4) = (3/4) 2 cups of chopped onion.


powfrac1


Notice that (3/4) 2 is a fraction raised to a power, namely 2, or a power of a fraction, and we want to look at how to simplify these powers of fractions.

In general, a power of a fraction is a fraction, called the base, raised to a number, called the exponent. In our example, 3/4 is the base, and 2 is the exponent.

Multiplying Fractions

Before we look at how to simplify powers of fractions, we need to be familiar with how to multiply fractions. Multiplying fractions is really quite simple. In fact, it's the easiest of all the basic operations done on fractions to perform. All we need to do is multiply the numerators to find the numerator of the product, and multiply the denominators to find the denominator of the product. This is illustrated in the following equation.


Rule for Multiplying Fractions
powfrac2


For example, if we wanted to multiply (2/7) * (5/6), we multiply the numerators to get 2*5 = 10, giving the numerator of the product. Then we multiply the denominators to get 7*6 = 42, giving the denominator of the product. Lastly, if possible, we simplify the result.


powfrac3


Simplifying Powers of Fractions

In our dinner example, we see that (3/4) 2 = (3/4) * (3/4), so let's carry this out.


powfrac4


This tells us that you want to add 9/16 of a cup of chopped onion to your dinner recipe.

Notice that to calculate (3/4) 2, we multiply 3/4 by itself two times. Also notice that the exponent is two. This is no coincidence! In general, when it comes to simplifying powers of fractions, we multiply the fraction by itself the number of times that the exponent indicates.

Let's consider a few more examples.


powfrac5


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