### 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*}

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*}

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*}

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*} 43 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*} 35**

#### 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*} 6×6×6**

#### 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*}