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# Whole Number Exponents

## Distinguish bases and powers.

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Whole Number Exponents

Sheila's parents are encouraging her to save her money. Sheila currently has 3. Sheila's parents tell her that for every month she saves her money instead of spending it, they will double the amount of money she has! Sheila decides she will save her money and not spend it for 6 months. How could Sheila write an expression in exponential form and evaluate to determine how much money she will have in 6 months? In this concept, you will learn how to write and evaluate expressions in exponential form. ### Writing and Evaluating Whole Number Exponents Sometimes you need to multiply a number or a variable by itself many times. Here is an example. \begin{align*}4 \times 4\times 4\times 4\times 4\times 4\times 4\end{align*} is 4 multiplied by itself 7 times. To avoid having to write out the 4 again and again, you can use an exponent. Whole number exponents are shorthand for repeated multiplication of a number by itself. \begin{align*}4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4=4^7\end{align*} In this example, 7 is the exponent and 4 is the base. The exponent indicates how many times the base is being multiplied by itself. Using an exponent can also be called “raising to a power. The exponent represents the power. For example, \begin{align*}4^7\end{align*} could be read 4 to the seventh power. There are two exponents that have special names. A base raised to the power of 2 is said to be squared. A base raised to the power of 3 is said to be cubed. Here is an example. • \begin{align*}4^2\end{align*} could be read 4 to the second power or 4 squared. • \begin{align*}4^3\end{align*} could be read 4 to the third power or 4 cubed. When you use an exponent to write an expression you are using exponential form. \begin{align*}4^7\end{align*} is exponential form. When you write out the expression using multiplication without an exponent you are using expanded form. \begin{align*}4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4\end{align*} is expanded form. Here is an example. Write the following in exponential form: \begin{align*}6 \times 6 \times 6 \times 6\end{align*} First, notice that 6 is being multiplied by itself 4 times. 6 will be your base and 4 will be your exponent. The answer is \begin{align*} 6 \times 6 \times 6 \times 6=6^4\end{align*}. Here is another example. Write the following in expanded form and evaluate the expression: \begin{align*}5^3\end{align*} First, read the expression. This expression could be read as 5 to the third power or 5 cubed. Next, write the expression in expanded form without an exponent. \begin{align*}5^3=5 \times 5 \times 5\end{align*} Then, multiply. \begin{align*}5 \times 5 \times 5=125\end{align*} The answer is \begin{align*}5^3=5 \times 5 \times 5=125\end{align*}. ### Examples #### Example 1 Earlier, you were given a problem about Sheila and her3.

Her parents told her they would double the amount of money she has for every month she saves her money instead of spending it. Sheila decides she will save her money for 6 months and wonders how much money she will have at this point.

First, write an expression to represent how much money Sheila will have after 6 months. Start with how much money she will have after one month and work your way up to 6 months. Remember that to double means to multiply by 2.

• Sheila starts with 3. • After 1 month she will have \begin{align*}\3 \times 2\end{align*}. • After 2 months she will have \begin{align*}\3 \times 2 \times 2\end{align*} or \begin{align*}\3 \times 2^2\end{align*}. • After 3 months she will have \begin{align*}\3 \times 2 \times 2 \times 2\end{align*} or \begin{align*}\3 \times 2^3\end{align*}. Continuing in this pattern you can see that • After 6 months she will have \begin{align*}\3 \times 2^6\end{align*}. Next, evaluate the expression in order to figure out how much money she will have in 6 months. First, write the expression in expanded form. \begin{align*}3 \times 2^6=3 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\end{align*} Now, multiply. \begin{align*}3 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2=192\end{align*} The answer is that after 6 months of saving Sheila will have192. Not bad!

#### Example 2

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

First, write the expression in expanded form.

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

Next, multiply each part of the expression.

\begin{align*}2 \times 2 \times 2+4 \times 4=8+16\end{align*}

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

The answer is \begin{align*}2^3+4^2=24\end{align*}.

#### Example 3

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

First, remember that to write an expression in exponential form you need a base and an exponent. The base is the number that is being multiplied by itself. The exponent is the number of times the base is being multiplied by itself.

The base is 3.

The exponent is 5.

Now, write in exponential form.

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

The answer is \begin{align*}3 \times 3 \times 3 \times 3 \times 3=3^5\end{align*}.

#### Example 4

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

First, write the expression in expanded form without an exponent.

\begin{align*}6^3=6 \times 6 \times 6\end{align*}

Next, multiply.

\begin{align*}6 \times 6 \times 6=216\end{align*}

The answer is \begin{align*}6^3=6 \times 6 \times 6=216\end{align*}.

#### Example 5

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

First, write the expression in expanded form.

\begin{align*}4^3-5^2=4 \times 4 \times 4-5 \times 5\end{align*}

Next, multiply each part of the expression.

\begin{align*}4 \times 4 \times 4-5 \times 5=64-25\end{align*}

Then, subtract.

\begin{align*}64-25=39\end{align*}

The answer is \begin{align*}4^3-5^2=39\end{align*}.

### Review

Name the base and exponent in the following expressions. 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*}2^4\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*}

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### Vocabulary Language: English

TermDefinition
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.
Expanded Form Expanded form refers to a base and an exponent written as repeated multiplication.
Exponent Exponents are used to describe the number of times that a term is multiplied by itself.
Exponential Notation Exponential notation is a way to write repeated multiplication using a base and an exponent.