1.3: Powers and Exponents
Introduction
The Tiger
Miguel is one of the designers at the city zoo where Jonah and Sarah have been spending the summer. He is working on the new tiger habitat.
Today while he is working on rebuilding part of the habitat, he has to move Leonard, a beautiful Bengal tiger, to one of the cages.
A tiger needs to have a cage that is a specific size so that he can pace and have enough room to not feel confined. If you have ever been to a zoo, you know that tigers LOVE to pace.
There are two cages for Miguel to choose from.
One has the dimensions \begin{align*}9^3\end{align*} feet.
The other has the dimensions \begin{align*}12^3\end{align*} feet.
A tiger’s cage must be 1728 cubic feet so that he can have enough room to pace.
Which cage has the right dimensions?
Is there one that will give Leonard more room to roam?
How can you compare the sizes of the cages?
In this lesson, you will learn how to use exponents to help Miguel select the correct cage for Leonard.
Pay close attention and we will solve this problem at the end of the lesson.
What You Will Learn
In this lesson, you will learn to:
- Distinguish between a whole number, a power, a base and an exponent
- Write the product of a repeating factor as a power
- Find the value of a number raised to a power
- Compare values of different bases and exponents
- Solve real-world questions using whole number powers
Teaching Time
I. Whole Numbers, Powers, Bases and Exponents
In the past two lessons you have been working with whole numbers.
A whole number is just that. It is a number that represents a whole quantity.
Today, we are going to learn about how to use exponents.
An exponent is a little number that is added to a whole number, but exponents are very powerful "little numbers". They change the meaning of the whole number as soon as they are added.
Here is an example.
The large number is called the base.
You can think about the base as the number that you are working with.
The small number is called the exponent.
The exponent tells us how many times to multiply the base by itself.
An exponent can also be known as a power.
We can read bases and exponents.
Here are some examples of how to read them.
\begin{align*}3^5\end{align*} is read as "three to the fifth power".
\begin{align*}2^7\end{align*} is read as "two to the seventh power".
\begin{align*}5^9\end{align*} is read as "five to the ninth power".
We could go on and on.
When you see a base with an exponent of 2 or an exponent of 3, we have different names for those.
We read them differently.
\begin{align*}2^2\end{align*} is read as two squared.
\begin{align*}6^3\end{align*} is read as six cubed.
It doesn’t matter what the base is, the exponents two and three are read squared and cubed.
What does an exponent actually do?
An exponent tells us how many times the base should be multiplied by itself.
We can write them out the long way.
Example
\begin{align*}7^3 = 7 \times 7 \times 7\end{align*}
Example
\begin{align*}5^{10} = 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5\end{align*}
If you haven’t figured it out yet, exponents are a multiplication short cut a lot like the way that multiplication is an addition short cut.
Here are few for you to work on by yourself.
- Write out in words - \begin{align*}6^3\end{align*}
- Write out the factors of \begin{align*}4^5\end{align*}
- Which is the base number: \begin{align*}9^{10}\end{align*}?
Take a minute and check your work with a peer.
II. Writing the Product of a Repeating Factor as a Power
In the last section, we took bases with exponents and wrote them out as factors.
We can also work the other way around.
We can take repeated factors and rewrite them as a power using an exponent.
Example
\begin{align*}7 \times 7 \times 7 = \underline{\;\;\;\;\;\;\;}\end{align*}
There are three seven’s being multiplied.
We rewrite this as a base with an exponent.
Example
\begin{align*}7 \times 7 \times 7 = 7^3\end{align*}
Example
\begin{align*}11 \times 11 \times 11 \times 11 = 11^4\end{align*}
III. Evaluating Powers
We can also find the value of a power by evaluating it.
This means that we actually complete the multiplication and figure out the new product.
Let’s look at an example.
Example
\begin{align*}5^2\end{align*}
We want to evaluate 5 squared. We know that this means \begin{align*}5 \times 5.\end{align*}
First, we write it out as factors.
Example
\begin{align*}5^2 = 5 \times 5\end{align*}
Next, we solve it.
Example
\begin{align*}5^2 = 5 \times 5 = 25 \end{align*}
RED ALERT!!! The most common mistake students make with exponents is to just multiply the base by the exponent.
\begin{align*}5^2\end{align*} IS NOT \begin{align*}5 \times 2\end{align*}
The exponent tells us how many times to multiply the base by itself.
\begin{align*}5^2\end{align*} is \begin{align*}5 \times 5\end{align*}
Be sure to keep this in mind!!!
Here are a few for you to evaluate on your own.
- \begin{align*}2^6\end{align*}
- \begin{align*}6^3\end{align*}
- \begin{align*}1^{100}\end{align*}
Take a few minutes and check your answers with a peer.
IV. Comparing Values of Powers
We can also compare the values of powers using greater than, less than and equal to.
We use our symbols to do this.
Greater than \begin{align*}>\end{align*}
Less than \begin{align*}<\end{align*}
Equal to \begin{align*}=\end{align*}
To compare the value of different powers, we will need to evaluate each power and then compare them.
Here is an example.
Example
\begin{align*}5^3 \underline{\;\;\;\;\;\;\;}6^2\end{align*}
First, we evaluate 5 cubed. \begin{align*}5^3 = 125\end{align*}
Next, we evaluate 6 squared. \begin{align*}6^2 = 36\end{align*}
Let’s rewrite the problem.
Example
One hundred and twenty-five is greater than thirty-six.
Here are a few for you to work through on your own.
- \begin{align*}2^7 \underline{\;\;\;\;\;\;\;}5^3\end{align*}
- \begin{align*}1^9 \underline{\;\;\;\;\;\;\;} 1^{14}\end{align*}
- \begin{align*}4^5 \underline{\;\;\;\;\;\;\;}5^4\end{align*}
Take a few minutes and check your work with a peer.
Real Life Example Completed
The Tiger
Having learned all about exponents and powers, you should be able to help Miguel with Leonard the Bengal tiger.
Let’s look back at the original dilemma.
Miguel is one of the designers at the city zoo where Jonah and Sarah have been spending the summer. He is working on the new tiger habitat.
Today while he is working on rebuilding part of the habitat, he has to move Leonard, a beautiful Bengal tiger, to one of the cages.
A tiger needs to have a cage that is a specific size so that he can pace and not feel confined. If you have ever been to a zoo, you know that tigers LOVE to pace.
There are two cages for Miguel to choose from.
One has the dimensions \begin{align*}9^3\end{align*} feet.
The other has the dimensions \begin{align*}12^3\end{align*} feet.
A tiger’s cage in a city zoo must be 1728 cubic feet.
Which cage has the right dimensions?
Is there one that will give Leonard more room to roam?
How can you compare the sizes of the cages?
First, let’s underline any information that seems important. This has been done for you in the paragraph above.
Our next step is to use what we learned about exponents and powers to evaluate the size of each cage.
The first cage has dimensions of \begin{align*}9^3\end{align*} feet.
We can evaluate that as \begin{align*}9 \times 9 \times 9 = 729 \ ft^3\end{align*}
Since we multiplied \begin{align*}\text{feet} \times \text{feet} \times \text{feet}\end{align*}, we write our answer as feet cubed, \begin{align*}ft^3\end{align*}. Therefore, the full answer is 729 \begin{align*}ft^3\end{align*}.
The second cage has dimensions of \begin{align*}12^3\end{align*} feet.
We can evaluate that as \begin{align*}12 \times 12 \times 12 = 1728 \ ft^3\end{align*}
We were given the fact that a tiger needs to have a cage that is 1728 cubic feet.
The second cage has the correct dimensions.
We can also compare the cage sizes using "greater than" or "less than" symbols.
\begin{align*}9^3 < 12^3\end{align*}
Miguel can now be confident that Leonard will have enough room to roam in his new cage.
Vocabulary
Here is the vocabulary that was used in this lesson. Remember, you can find these words in italics throughout the lesson.
- Whole number
- a number that represents a whole quantity
- Base
- the whole number part of a power
- Power
- the value of the exponent
- Exponent
- the little number that tells how many times we need to multiply the base by itself
- Squared
- the name used to refer to the exponent 2
- Cubed
- the name used to refer to the exponent 3
Technology Integration
Khan Academy Level 1 Exponents
James Sousa Examples of Exponents
http://got.im/Vzw - This website works on explaining how students can work with powers and exponents.
Time to Practice
Directions: Write each power out in words.
1. \begin{align*}3^2\end{align*}
2. \begin{align*}5^5\end{align*}
3. \begin{align*}6^3\end{align*}
4. \begin{align*}2^6\end{align*}
5. \begin{align*}7^2\end{align*}
Directions: Write each repeated factor using a power.
6. \begin{align*}4 \times 4 \times 4\end{align*}
7. \begin{align*}3 \times 3 \times 3 \times 3\end{align*}
8. \begin{align*}2 \times 2\end{align*}
9. \begin{align*}9 \times 9 \times 9 \times 9 \times 9\end{align*}
10. \begin{align*}10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10\end{align*}
11. \begin{align*}1 \times 1 \times 1 \times 1 \times 1 \times 1 \times 1 \times 1 \times 1 \times 1\end{align*}
12. \begin{align*}3 \times 3 \times 3 \times 3 \times 3 \times 3\end{align*}
13. \begin{align*}4 \times 4\end{align*}
14. \begin{align*}7 \times 7 \times 7\end{align*}
15. \begin{align*}20 \times 20 \times 20 \times 20\end{align*}
Directions: Evaluate the value of each power.
16. \begin{align*}2^2\end{align*}
17. \begin{align*}3^2\end{align*}
18. \begin{align*}6^2\end{align*}
19. \begin{align*}7^3\end{align*}
20. \begin{align*}8^4\end{align*}
21. \begin{align*}2^6\end{align*}
22. \begin{align*}3^5\end{align*}
23. \begin{align*}6^4\end{align*}
24. \begin{align*}5^3\end{align*}
25. \begin{align*}1^{100}\end{align*}
Directions: Compare each power using \begin{align*}<\end{align*}, \begin{align*}>\end{align*}, or \begin{align*}=\end{align*}
26. \begin{align*}4^2 \underline{\;\;\;\;\;\;\;}2^4\end{align*}
27. \begin{align*}3^2 \underline{\;\;\;\;\;\;\;}1^5\end{align*}
28. \begin{align*}6^3 \underline{\;\;\;\;\;\;\;}3^6\end{align*}
29. \begin{align*}7^2 \underline{\;\;\;\;\;\;\;} 5^2\end{align*}
30. \begin{align*}8^3 \underline{\;\;\;\;\;\;\;} 9^2\end{align*}