The Quotient Rule for Exponents

Instructor: Stephanie Matalone

Stephanie taught high school science and math and has a Master's Degree in Secondary Education.

This lesson will go over a very important rule in the exponent world called the quotient rule. We will discuss how we can use this rule to divide exponents, and go over examples of using this rule in different scenarios.

Introduction

You are responsible for handing out jellybeans at a huge party coming up. What a big responsibility! You have 1,000,000,000 jellybeans to pass out evenly to 10,000 people. You know you need to divide the numbers, but is there an easier way than using long division? YES! We can use exponents to solve this pesky jellybean conundrum.

1,000,000,000 is the same as 109 and 10,000 is the same as 104. We need to divide 109 by 104 to figure out how many jelly beans each person gets at the party.

The division problem looks like this:

Exponent Basics

The base is the large number that is being multiplied by itself, while the exponent is the small number that tells you how many times to multiply the base by itself. In the example below, 7 would be the base, while 6 would be the exponent.

This means that 7 would be multiplied by itself 6 times to equal 117,649.

Quotient Rule

Now, back to the jellybeans. To divide these two exponents with a base of ten, we will use the quotient rule. This rule states that when you are dividing two exponents with the same base, you must subtract the exponents. In our example, both the numerator and denominator (top and bottom of the division problem) have 10 as a base. This means we can subtract the exponents to simplify our problem.

We will start with the exponent of 9 from the numerator. Then we subtract the exponent of 4 from the denominator. Our final answer will be 105. This means that each person at the party will get 100,000 jellybeans! That's a lot of jellybeans per person. You are going to need some big gift bags.

Examples

Let's go over something you are more likely to see in the math world. This time, we will look at an example with a variable. A variable is a letter that stands for an unknown value. In this example, the variable will be the letter x.

We follow the same steps with the jellybean problem, except that this time, our base is x. This means our answer will be x(12-8), which simplifies to x4.

Let's now try one with multiple variables. This time we will have x, y, and z variables. Be aware that a y variable can only be divided by another y variable. Keep like with like!

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