### 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**

### 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

Here is a video for 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*}