1.3: Using Exponents
Introduction
Getting Ready
On the first day of Teen Adventure, Kelly thought they would be hiking, but when the group assembled at the Lafayette Place Campground she realized that there was a lot to do before they could begin hiking. First, the leaders organized each group into 10 hikers with 2 leaders each. Then the leaders split off with their groups to do some training.
There was a lot to learn. As the leaders of Kelly’s group, Scott and Laurel began by having the hikers introduce themselves and share a little about their hiking experience. The hikers learned that the group would be taking it easy the first week while everyone got into shape and had a chance to get to know each other. The hiking would get more strenuous as the time went on.
After introductions, Scott and Laurel gave the campers two tents. Since there were five boys and five girls in each group, the team would need two tents. There would be times when they would be sleeping in cabins, but there also would be times where tents would be necessary.
Their first task was to set up the tent and figure out the square footage of the floor. The girls and boys were each given a Kelty Trail Dome 6.
Kelly and the other girls took one tent and began to take it out of its package. They were so excited that they did not pay attention and almost lost the directions. Luckily, Kara saw this and caught them before the wind did. Kelly read the directions. The tent was sized to sleep six so it would be perfect for the 5 girls and one of the leaders.
Dimensions of the floor \begin{align*}= 120^2\end{align*} square inches
Kelly and Jessica looked at the dimensions. Who would have thought that they would be solving math problems when hiking! Jessica took out a piece of paper and began working on the problem.
\begin{align*}120^2\end{align*} square inches is a measurement that has an exponent. To figure out the dimensions of the floor of the tent Kelly and Jessica will need to know how to work with exponents. In this lesson you will learn all about exponents. By the end of this lesson, you will know how to help Kelly and Jessica figure out the area of the tent floor.
What You Will Learn
In this lesson you will learn the following skills:
- Identify whole number powers, bases and exponents.
- Evaluate powers with variable bases.
- Write variable expressions involving exponents to represent and solve real-world problems.
Teaching Time
I. Identify Whole Number Powers, Bases and Exponents
Sometimes, we have to multiply the same number several times. We can say that we are multiplying the number by itself in this case.
\begin{align*}4 \times 4 \times 4\end{align*} is 4 multiplied by itself three times.
When we have a situation like this, it is helpful to use a little number to show how many times to multiply the number by itself. That little number is called an exponent.
If we were going to write \begin{align*}4 \times 4 \times 4\end{align*} with an exponent, we would write \begin{align*}4^3\end{align*}. This lesson is all about exponents. By the end of it, you will how and when to use them and how helpful this shortcut is for multiplication.
Using exponents has an even more technical term also. We can say that we use exponential notation when we express multiplication in terms of exponents.
We use exponential notation to write an expanded multiplication problem as a base number with an exponent, we write \begin{align*}4 \times 4 \times 4\end{align*} with an exponent \begin{align*}= 4^3\end{align*}
We can work the other way around too. We can write a number with an exponent as a long multiplication problem and this is called expanded form.
The base number is the number being multiplied by itself; in this case the base is 4.
The exponent tells how many times to multiply the base by itself; in this case it is 3.
Using an exponent can also be called “raising to a power.” The exponent represents the power.
Here \begin{align*}4^3\end{align*} would be read as “Four to the third power.”
Let’s look at an example.
Example
Write the following in exponential notation: \begin{align*}6 \times 6 \times 6 \times 6\end{align*}
Exponential Notation means to write this as a base with an exponent.
Six multiplied by itself four times \begin{align*}= 6^4\end{align*}
This is our answer.
Example
Write the following in expanded form: \begin{align*}5^3\end{align*}
Expanded form means to write this out as a multiplication problem.
\begin{align*}5 \times 5 \times 5\end{align*}
This is our answer.
We can also evaluate expressions to find a single value.
Example
\begin{align*}4^3\end{align*}
Our first step is to write it out into expanded form.
\begin{align*}4 \times 4 \times 4\end{align*}
Now multiply.
\begin{align*}4 \times 4 = 16 \times 4 = 64\end{align*}
Our answer is 64.
Here is one more that is a little harder. It is an example that is an expression with two terms.
Example
\begin{align*}2^3+4^2\end{align*}
To evaluate this expression, write it out in expanded form.
\begin{align*}(2)(2)(2) + (4)(4)\end{align*}
Now multiply each part of the expression and add the results.
\begin{align*}8 + 16\!\\ 24\end{align*}
Our answer is 24.
1G. Lesson Exercises
- Write the following in exponential notation: \begin{align*}3 \times 3 \times 3 \times 3 \times 3\end{align*}
- Write in expanded form and then evaluate the expression: \begin{align*}6^3\end{align*}
- Evaluate: \begin{align*}4^3-5^2\end{align*}
Take a few minutes to check your work with a partner.
II. Evaluate Powers with Variable Bases
When we are dealing with numbers, it is often easier to just simplify. It makes more sense to deal with 16 than with \begin{align*}4^2\end{align*}. Exponential notation really comes in handy when we’re dealing with variables. It is easier to write \begin{align*}y^{12}\end{align*} than it is to write \begin{align*}yyyyyyyyyyyy\end{align*}.
We can simplify by using exponential form and we can also write out the variable expression by using expanded form (repeated multiplication).
Example
Write the following in expanded form: \begin{align*}x^5\end{align*}
To write this out, we simply write \begin{align*}x\end{align*} five times.
\begin{align*}x^5=xxxxx\end{align*}
We can work the other way too by taking an variable expression in expanded form using repeated multiplication and write it in exponential form.
Example
\begin{align*}aaaa\end{align*}
Our answer is \begin{align*}a^4\end{align*}.
What about when we multiply two variable terms with exponents?
To do this, we are going to need to follow a few rules. Let’s look at an example and then work through it.
Example
\begin{align*}(m^3)(m^2)\end{align*}
The first thing to notice is that these terms have the same base. Both bases are m’s. Because of this, we can simplify the expression quite easily.
Let’s write it out in expanded form.
\begin{align*}mmm(mm)\end{align*}
Here we have five \begin{align*}m\end{align*}’s being multiplied, so our answer is \begin{align*}m^5\end{align*}.
Here is the rule.
Let’s apply this rule to the next example.
Example
\begin{align*}(x^6)(x^3)\end{align*}
The bases are the same, so we add the exponents.
\begin{align*}x^{6+3}= x^9\end{align*}
This is the answer.
In these examples we multiplied two exponential terms. We can also have an exponential term raised to a power. When this happens, one exponent is outside the parentheses. This means something different. Take a look at this example.
Example
\begin{align*}(x^2)^3\end{align*}
Let’s think about what this means. It means that we are multiplying \begin{align*}x\end{align*} squared by itself three times. We can write this out in expanded form as:
\begin{align*}(x^2)(x^2)(x^2)\end{align*}
Now we are multiplying three bases that are the same, so we use Rule 1 and add the exponents.
Our answer is \begin{align*}x^6\end{align*}.
We could have multiplied the two exponents in the beginning.
\begin{align*}(x^2)^3= x^{2(3)} =x^6\end{align*}
This is Rule 2.
Example
Simplify \begin{align*}x^0\end{align*}
Our answer is \begin{align*}x^0 = 1\end{align*}
Anything to the power of 0 equals 1.
Take a few notes on the rules before moving on in the lesson.
1H. Lesson Exercises
- Write the following in exponential form: \begin{align*}aaaaaaa\end{align*}
- Simplify: \begin{align*}(a^3)(a^8)\end{align*}
- Simplify: \begin{align*}(x^4)^2\end{align*}
Check your answers with a friend.
III. Write Variable Expressions Involving Exponents to Represent and Solve Real-World Problems
The hikers in the beginning of the lesson were learning to use exponents in figuring out their tent dimensions. We can also look at a problem with swimmers to see how exponents are featured in real life examples.
Example
Jessica swam four miles during the first week of swim camp. Every week thereafter, she increased the number of miles that she swam by four times. How many miles did Jessica swim during the fourth week?
To work through this problem let’s create a list of mileage.
Week \begin{align*}1 = 4 \ \text{miles}\end{align*}
Week \begin{align*}2 = 4 \times 4\end{align*}
Week \begin{align*}3 = 4 \times 4 \times 4\end{align*}
Week \begin{align*}4 = 4 \times 4 \times 4 \times 4\end{align*}
We can use exponents to show Jessica’s mileage for week 4.
\begin{align*}4 \times 4 \times 4 \times 4 = 4^4\end{align*}
Jessica swam 256 miles during the fourth week.
Real Life Example Completed
Getting Ready
Here is the original problem once again. Reread the problem and underline any important information before beginning.
On the first day of Teen Adventure, Kelly thought they would be hiking, but when the group assembled at the Lafayette Place Campground she realized that there was a lot to do before they could begin hiking. First, the leaders organized each group into 10 hikers with 2 leaders each. Then the leaders split off with their groups to do some training.
There was a lot to learn. As the leaders of Kelly’s group, Scott and Laurel began by having the hikers introduce themselves and share a little about their hiking experience. The hikers learned that the group would be taking it easy the first week while everyone got into shape and had a chance to get to know each other. The hiking would get more strenuous as the time went on.
After introductions, Scott and Laurel gave the campers two tents. Since there were five boys and five girls in each group, the team would need two tents. There would be times when they would be sleeping in cabins, but there also would be times where tents would be necessary.
Their first task was to set up the tent and figure out the square footage of the floor. The girls and boys were each given a Kelty Trail Dome 6.
Kelly and the other girls took one tent and began to take it out of its package. They were so excited that they did not pay attention and almost lost the directions. Luckily, Kara saw this and caught them before the wind did. Kelly read the directions. The tent was sized to sleep six so it would be perfect for the 5 girls and one of the leaders.
Dimensions of the floor \begin{align*}= 120^2\end{align*} square inches
Kelly and Jessica looked at the dimensions. Who would have thought that they would be solving math problems when hiking! Jessica took out a piece of paper and began working on the problem.
First, notice that the measurement is in square inches, not square feet. Our final answer needs to be in square footage, so while figuring out these dimensions, the girls will need to convert the measurement to feet. The area of a square is one place where we use exponents all the time. The length of one side of a square is called \begin{align*}s\end{align*}, so we can write \begin{align*}s^2\end{align*} to find the area of a square. Since the tent floor is square, the dimensions have been written in square inches.
Since the girls need to find the floor area in square feet, instead of square inches, the decide to first convert each dimension from inches to feet. Since there are 12 inches in 1 foot, the girls divide each side by 12 and come up with:
120 inches divided by 12 equals 10 feet
Now they use the new information to multiply the dimensions of each side and get:
10 feet times 10 feet (or 10 feet squared) = 100 square feet
Exponents are very useful when working with area!
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Exponent
- a little number that tells you how many times to multiply the base by itself.
- Base
- the big number in a variable expression with an exponent.
- Exponential Notation
- writing multiplication using a base and an exponent
- Expanded Form
- Removing the exponent from a base and writing out the expression using repeated multiplication.
Technology Integration
Khan Academy Exponent Rules Part 1
Just Math Tutoring, Working with Exponents
James Sousa, Exponential Notation
James Sousa, Example of Writing Repeated Multiplication Using Exponents
James Sousa, Expanding and Evaluating Exponential Notation
James Sousa, Simplifying Exponential Expressions - Product and Power Rules
Time to Practice
Directions: Name the base and exponent in the following examples. Then write in expanded form and write how to “read” the exponential notation, for example: "four to the fifth power".
1. \begin{align*}4^5\end{align*}
2. \begin{align*}3^2\end{align*}
3. \begin{align*}5^8\end{align*}
4. \begin{align*}4^3\end{align*}
5. \begin{align*}6^3\end{align*}
6. \begin{align*}2^5\end{align*}
7. \begin{align*}1^{10}\end{align*}
Directions: Evaluate each expression.
8. \begin{align*}2^3\end{align*}
9. \begin{align*}4^2\end{align*}
10. \begin{align*}5^2\end{align*}
11. \begin{align*}9^0\end{align*}
12. \begin{align*}5^3\end{align*}
13. \begin{align*}2^6\end{align*}
14. \begin{align*}3^3\end{align*}
15. \begin{align*}3^2+4^2\end{align*}
16. \begin{align*}5^3+2^2\end{align*}
17. \begin{align*}6^2+2^3\end{align*}
18. \begin{align*}6^2-5^2\end{align*}
19. \begin{align*}2^4-2^2\end{align*}
20. \begin{align*}7^2+3^3+2^2\end{align*}
Directions: Simplify the following variable expressions.
21. \begin{align*}(m^2)(m^5)\end{align*}
22. \begin{align*}(x^3)(x^4)\end{align*}
23. \begin{align*}(y^5 )(y^3)\end{align*}
24. \begin{align*}(b^7 )(b^2)\end{align*}
25. \begin{align*}(a^5 )(a^2)\end{align*}
26. \begin{align*}(x^9 )(x^3)\end{align*}
27. \begin{align*}(y^4 )(y^5)\end{align*}
Directions: Simplify.
28. \begin{align*}(x^2 )^4\end{align*}
29. \begin{align*}(y^5 )^3\end{align*}
30. \begin{align*}(a^5 )^4\end{align*}
31. \begin{align*}(x^2 )^8\end{align*}
32. \begin{align*}(b^3 )^4\end{align*}