Have you ever been hiking with a group of other teens?

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. The leaders of Kelly’s group, Scott and Laurel began by having the hikers introduce themselves and share a little about their hiking experience. They 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 looked at the directions. The tent was sized to sleep six so it would be perfect for them and one of the leaders.

Dimensions of the floor \begin{align*}= 120^2\end{align*} 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*} **inches is a measurement that has an exponent. To figure out the dimensions of the floor of the tent you will need to know how to work with exponents. In this Concept, you will learn all about exponents. By the end, you will know how to figure out the area of floor of the tent.**

### Guidance

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 fancier name too. We can say that we use ** exponential notation** when we express multiplication in terms of exponents.

We can use ** exponential notation** to write an expanded multiplication problem into a form 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** 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 a 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.”**

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 times itself four times** \begin{align*}= 6^4\end{align*}

**This is our answer.**

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 with variables.**

\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.**

Now it's time for you to try a few on your own.

#### Example A

Write the following in exponential form: \begin{align*}3 \times 3 \times 3 \times 3 \times 3\end{align*}

**Solution: \begin{align*}3^5\end{align*}**

#### Example B

Write the following in expanded form and evaluate the expression: \begin{align*}6^3\end{align*}

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

#### Example C

Evaluate: \begin{align*}4^3-5^2\end{align*}

**Solution: 39**

Now let's go back to the tent dilemma from the beginning of the Concept.

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 looked at the directions. The tent was sized to sleep six so it would be perfect for them and one of the leaders.

Dimensions of the floor \begin{align*}= 120^2\end{align*} 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 inches not feet. Our final answer needs to be in square footage, so after 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 square has side \begin{align*}x\end{align*} side, 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.

\begin{align*}120^2\end{align*} **inches**

To start, the girls need to multiply this out.

\begin{align*}120 \times 120\end{align*}

Next, they can covert each inch dimension to feet.

There are 12 inches in 1 foot, so we divide each measurement by 12. 120 divided by \begin{align*}12 = 10\end{align*}.

Now we multiply to find the area in square feet.

\begin{align*}10 \ ft \times 10 \ ft = 100\end{align*} **square feet**

**Exponents are very useful when working with area!**

### Vocabulary

- 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 long multiplication using a base and an exponent

- Expanded Form
- taking a base and an exponent and writing it out as a long multiplication problem.

### Guided Practice

Here is one for you to try on your own.

\begin{align*}2^3+4^2\end{align*}

**Answer**

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.

\begin{align*}& 8 + 16\\ & 24\end{align*}

**Our answer is 24.**

### Video Review

- This is a James Sousa video about writing numbers in exponential form.

### Practice

Directions: Name the base and exponent in the following examples. Then write each in expanded form.

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*}

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

9. \begin{align*}3^{4}\end{align*}

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

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

12. \begin{align*}8^{10}\end{align*}

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

14. \begin{align*}12^{2}\end{align*}

15. \begin{align*}13^{3}\end{align*}