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# 1.7: Whole Number Exponents

Difficulty Level: At Grade Created by: CK-12
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Have you ever designed a tiger cage? Do you know how to use exponents to solve real world problems? Look at what Miguel learned about this very topic.

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 93\begin{align*}9^3\end{align*} feet.

The other has the dimensions 123\begin{align*}12^3\end{align*} feet.

A tiger’s cage in a city zoo should 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?

In this Concept, 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 Concept.

### Guidance

A whole number 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.

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.

35\begin{align*}3^5\end{align*} is read as "three to the fifth power".

27\begin{align*}2^7\end{align*} is read as "two to the seventh power".

59\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.

22\begin{align*}2^2\end{align*} is read as two squared.

63\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.

73=7×7×7

510=5×5×5×5×5×5×5×5×5×5

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 problems for you to try on your own.

#### Example A

Write out in words 63\begin{align*}6^3\end{align*}

Solution: 6^3 = 6 times 6 times 6

#### Example B

Write out the factors of 45\begin{align*}4^5\end{align*}

Solution:

45=4×4×4×4×4

#### Example C

Which is the base number: 910\begin{align*}9^{10}\end{align*}?

Solution: 9

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 93\begin{align*}9^3\end{align*} feet. The other has the dimensions 123\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 93\begin{align*}9^3\end{align*} feet.

We can evaluate that as 9×9×9=729 ft3\begin{align*}9 \times 9 \times 9 = 729 \ ft^3\end{align*}

Since we multiplied feet×feet×feet\begin{align*}\text{feet} \times \text{feet} \times \text{feet}\end{align*}, we write our answer as feet cubed, ft3\begin{align*}ft^3\end{align*}. Therefore, the full answer is 729 ft3\begin{align*}ft^3\end{align*}.

The second cage has dimensions of 123\begin{align*}12^3\end{align*} feet. We can evaluate that as 12×12×12=1728 ft3\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.

93<123

Miguel can now be confident that Leonard will have enough room to roam in his new cage.

### Vocabulary

Here are the vocabulary words that were used in this Concept.

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

### Guided Practice

Here's one for you to try on your own.

Write out the factors of 35\begin{align*}3^5\end{align*}

Then evaluate the product.

35=3×3×3×3×3

243

### Practice

Directions: Write each power out in words.

1. 32\begin{align*}3^2\end{align*}

2. 55\begin{align*}5^5\end{align*}

3. 63\begin{align*}6^3\end{align*}

4. 26\begin{align*}2^6\end{align*}

5. 34\begin{align*}3^4\end{align*}

6. 74\begin{align*}7^4\end{align*}

7. 52\begin{align*}5^2\end{align*}

8. 24\begin{align*}2^4\end{align*}

9. 33\begin{align*}3^3\end{align*}

10. 93\begin{align*}9^3\end{align*}

Directions: Now evaluate each power in problems 1 - 10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

Oct 29, 2012

Jul 08, 2015