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# 1.8: Values Written as Powers

Difficulty Level: At Grade Created by: CK-12
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Practice Values Written as Powers

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Remember Miguel and the tiger cage? Well, while he was working on his design, he also went and visited a friend in another zoo.This zoo also had a tiger cage, and it had dimensions for height, length and width just like the one that Miguel was working to design.

The tiger cage at this zoo had a width of 18 feet, a height of 18 feet and a length of 18 feet. Miguel wrote down 18 x 18 x 18 in his notebook.

There is an easier way to write this though. In this Concept, you will learn how to write the product of repeated factors by using powers. Pay attention and you will be able to do this by end of the Concept.

### Guidance

In the last Concept, 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.

To work in this way, we will count the number of times the base is being multiplied. This becomes our exponent. Remember that an exponent is the little number that tells you how many times to multiply the base by itself.

\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.

\begin{align*}7 \times 7 \times 7 = 7^3\end{align*}

Now let's practice with a few examples.

#### Example A

\begin{align*}6 \times 6 \times 6 \times 6\end{align*}

Solution: \begin{align*}6^4\end{align*}

#### Example B

\begin{align*}2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\end{align*}

Solution: \begin{align*}2^7\end{align*}

#### Example C

\begin{align*}3 \times 3\end{align*}

Solution: \begin{align*}3^2\end{align*}

Now let's go back to the original problem about the dimensions of the tiger cage.

Miguel wrote down 18 feet x 18 feet x 18 feet. Because these are repeated factors being multiplied, Miguel can use a short - cut and use an exponent to express the repeated multiplication. 18 is the base. Because it is being multiplied three times, three is the exponent.

The solution is \begin{align*}18^3\end{align*}.

### Vocabulary

Here are the vocabulary words 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.

\begin{align*}4 \times 4 \times 4 \times 4\end{align*}

\begin{align*}4^4\end{align*}

### Practice

Directions: Write each repeated factor using a power.

1. \begin{align*}4 \times 4 \times 4\end{align*}

2. \begin{align*}3 \times 3 \times 3 \times 3\end{align*}

3. \begin{align*}2 \times 2\end{align*}

4. \begin{align*}9 \times 9 \times 9 \times 9 \times 9\end{align*}

5. \begin{align*}10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10\end{align*}

6. \begin{align*}1 \times 1 \times 1 \times 1 \times 1 \times 1 \times 1 \times 1 \times 1 \times 1\end{align*}

7. \begin{align*}3 \times 3 \times 3 \times 3 \times 3 \times 3\end{align*}

8. \begin{align*}4 \times 4\end{align*}

9. \begin{align*}7 \times 7 \times 7\end{align*}

10. \begin{align*}6 \times 6 \times 6 \times 6\end{align*}

11. \begin{align*}11 \times 11 \times 11\end{align*}

12. \begin{align*}12 \times 12\end{align*}

13. \begin{align*}18 \times 18 \times 18\end{align*}

14. \begin{align*}21 \times 21 \times 21 \times 21\end{align*}

15. \begin{align*}17 \times 17\end{align*}

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

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

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

Exponent

Exponents are used to describe the number of times that a term is multiplied by itself.

Power

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

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

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

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