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# Whole Number Exponents

## Distinguish bases and powers.

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Whole Number Exponents

### Let's Think About It

Credit: Seabamirum
Source: https://www.flickr.com/photos/seabamirum/2850890725
License: CC BY-NC 3.0

Horatio is a carpenter building a habitat for tigers at his local zoo. He has been told that the tiger’s enclosure needs to be at least 1720 cubic feet. Horatio has only ever built two animal habitats before, and he was hoping to use one of those as a model. One habitat was 93\begin{align*}9^3\end{align*} cubic feet, and the other was 123\begin{align*}12^3\end{align*} cubic feet. Which of Horatio’s previous habitat designs meets the requirements for the tiger habitat?

In this concept, you will learn how powers, bases, and exponents work together.

### Guidance

An exponent is a little number that shows how many times to multiply a certain number by itself.

Let's look at an example:  35\begin{align*}3^5\end{align*}

The large number, 4, is called the base

The small number, 2, is called the exponent. This number tells you how many times to multiply the base by itself.

When a base has an exponent of 2, such as 42\begin{align*}{4}^{2}\end{align*}, you read it as "four squared".

When a base has an exponent of 3, such as 63\begin{align*}{6}^{3}\end{align*}, you read it as "six cubed".

For all other exponents, you read the exponent as a power. For example,

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".

An exponent tells you how many times the base should be multiplied by itself. If you look at a number with an exponent written out the long way, you can see why exponents are useful.

7×7×7=73

In this case, there are three factors of 7. Remember that factors are numbers or variables multiplied by each other.

5×5×5×5×5×5×5×5×5×5=510

In this case, there are ten factors of 5.

Exponents are a multiplication shortcut a lot like the way that multiplication is an addition shortcut.

### Guided Practice

Write out the factors of 35\begin{align*}3^5\end{align*}, then evaluate the product.

35=3×3×3×3×3

Once you have written out all the factors, simply multiply from left to right.

3×3=9×3=27×3=81×3=243

The answer is 243.

### Examples

#### Example 1

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

First, identify the base number, which is the big number that will be multiplied by itself. In this case, the base is 6.

Next, consider the power, which is the small number that describes how many times the number will be multiplied. In this case, the power is 3, so the base will be multiplied by itself 3 times.

The solution is "6 times 6 times 6."

#### Example 2

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

First, remember that factors are numbers that are multiplied by each other to equal a certain value.

Next, identify the base number, which is the big number that will be multiplied by itself. In this case, the base is 4.

Then consider the power, which is the small number that describes how many times the number will be multiplied. In this case the power is 5, so the base will be multiplied by itself 5 times.

The solution is  45=4×4×4×4×4\begin{align*}4^5 = 4 \times 4 \times 4 \times 4 \times 4\end{align*}.

#### Example 3

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

First, remember that the base is the big number that is multiplied by itself.

Then, remember that the power is the little number that describes how many times to multiply the base by itself.

The answer is that the base in the expression 910\begin{align*}9^{10}\end{align*} is 9 and the power is 10.

Credit: Cliff1066
Source: https://www.flickr.com/photos/nostri-imago/2854166745
License: CC BY-NC 3.0

Remember Horatio and his tiger housing project?

Horatio needs to figure out which dimensions are closest to 1720 cubic feet: 93 cubic feet or 123 cubic feet.

First, write out the factors for each expression.

93=9×9×9

123=12×12×12

Next, multiply.

9×912×12=81×9=729=144×12=1728

The answer is 123.

Horatio can use his old 123 cubic feet habitat design, which even gives the tigers a little extra room.

### Explore More

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*}

Evaluate each expression (they are the same expressions as in problems 1 - 10).

11. 32\begin{align*}3^2\end{align*}

12. 55\begin{align*}5^5\end{align*}

13. 63\begin{align*}6^3\end{align*}

14. 26\begin{align*}2^6\end{align*}

15. 34\begin{align*}3^4\end{align*}

16. 74\begin{align*}7^4\end{align*}

17. 52\begin{align*}5^2\end{align*}

18. 24\begin{align*}2^4\end{align*}

19. 33\begin{align*}3^3\end{align*}

20. 93\begin{align*}9^3\end{align*}

### Vocabulary Language: English

Base

Base

When a value is raised to a power, the value is referred to as the base, and the power is called the exponent. In the expression $32^4$, 32 is the base, and 4 is the exponent.
Cubed

Cubed

The cube of a number is the number multiplied by itself three times. For example, "two-cubed" = $2^3 = 2 \times 2 \times 2 = 8$.
Power

Power

The "power" refers to the value of the exponent. For example, $3^4$ is "three to the fourth power".
Squared

Squared

Squared is the word used to refer to the exponent 2. For example, $5^2$ could be read as "5 squared". When a number is squared, the number is multiplied by itself.
Whole Numbers

Whole Numbers

The whole numbers are all positive counting numbers and zero. The whole numbers are 0, 1, 2, 3, ...

1. [1]^ Credit: Seabamirum; Source: https://www.flickr.com/photos/seabamirum/2850890725; License: CC BY-NC 3.0
2. [2]^ Credit: Cliff1066; Source: https://www.flickr.com/photos/nostri-imago/2854166745; License: CC BY-NC 3.0

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