Cherry is training to be an interior designer. One of her projects is to make notes on every room in which she spends time. Cherry soon gets sick of writing down all the dimensions: 18 feet wide x 18 feet long x 18 feet high. Is there an easier way for Cherry to note down all the same information?

In this concept, you will learn how to write the product of repeated values using powers.

### Exponents

**Factors** are a sequence of numbers that are multiplied by each other, like , where 2, 3, and 4 are factors, or \begin{align*}5 \times 5 \times 5 \times 5\end{align*}, where 5, 5, 5, and 5 are factors.

Repeated factors can be rewritten as a power using an **exponent. **Remember, the **exponent** is the little number that tells you how many times to multiply the base by itself.

Consider this expression:

\begin{align*}7 \times 7 \times 7 = \underline{\;\;\;\;\;\;\;}\end{align*}

There are three 7’s being multiplied. Since 7 is the number being multiplied, the base is 7. Since the 7 is multiplied by itself 3 times, the exponent (also called the power) is 3.

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

### Examples

#### Example 1

Earlier, you were given a problem about Cherry and her lengthy dimensions.

Cherry wants to know an easier alternative to writing 18 feet wide x 18 feet long x 18 feet high.

You now know that when the same number, or factor, is being multiplied repeatedly, you can represent that with an exponent.

In Cherry’s case, the base number is 18, and she is multiplying it 3 times, so 3 is the exponent. There is also a particular way to express the exponent 3: you can say the number is "cubed". In this case, since the units, feet, are multiplied by themselves three times also, the resulting units are "feet cubed" or "cubic feet".

The answer is .

Cherry can now make her notations much faster by writing 18^{3} ft^{3}, or 18 cubed, cubic feet.

#### Example 2

Write the following as a base with an exponent.

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

First, look at what number is being multiplied by itself, in this case, 4. That number is the base.

Next, write the base, 4, as a full-size number.

Then, write the power (also 4 in this case, since the base is repeated 4 times), as a small number above and to the right of the base.

The answer is \begin{align*}4^4\end{align*}.

#### Example 3

Write the following as a base with an exponent.

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

First, look at what number is being multiplied by itself. In this case, 6. This is the base.

Next, write the base, 6, as a full-size number.

Then, write the power, 4 in this case, as a small number above and to the right of the base.

The answer is \begin{align*}6^4\end{align*}.

#### Example 4

Write the following as a base with an exponent.

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

First, look at what number is being multiplied by itself. In this case, 2. That means 2 is the base.

Next, write the base, 2, as a full-size number.

Then, write the power, 7 in this case, as a small number above and to the right of the base.

The solution is

.#### Example 5

Write the following as a base with an exponent.

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

First, look at what number is being multiplied by itself. In this case, 3. This is the base.

Next, write the base, 3, as a full-size number.

Then, write the power, 2 in this case, as a small number above and to the right of the base.

The answer is

.### Review

Write each repeated factor using an exponent.

- \begin{align*}4 \times 4 \times 4\end{align*}
- \begin{align*}3 \times 3 \times 3 \times 3\end{align*}
- \begin{align*}2 \times 2\end{align*}
- \begin{align*}9 \times 9 \times 9 \times 9 \times 9\end{align*}
- \begin{align*}10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10\end{align*}
- \begin{align*}1 \times 1 \times 1 \times 1 \times 1 \times 1 \times 1 \times 1 \times 1 \times 1\end{align*}
- \begin{align*}3 \times 3 \times 3 \times 3 \times 3 \times 3\end{align*}
- \begin{align*}4 \times 4\end{align*}
- \begin{align*}7 \times 7 \times 7\end{align*}
- \begin{align*}6 \times 6 \times 6 \times 6\end{align*}
- \begin{align*}11 \times 11 \times 11\end{align*}
- \begin{align*}12 \times 12\end{align*}
- \begin{align*}18 \times 18 \times 18\end{align*}
- \begin{align*}21 \times 21 \times 21 \times 21\end{align*}
- \begin{align*}17 \times 17\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.8.