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Whole Number Exponents

Distinguish bases and powers.

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Whole Number Exponents
License: CC BY-NC 3.0

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.

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*}4×4×4×4×4×4×4 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*}4×4×4×4×4×4×4=47

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*}47 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*}42 could be read 4 to the second power or 4 squared.
  • \begin{align*}4^3\end{align*}43 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*}47 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*}4×4×4×4×4×4×4 is expanded form.

Here is an example.

Write the following in exponential form: \begin{align*}6 \times 6 \times 6 \times 6\end{align*}6×6×6×6

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*}6×6×6×6=64.

Here is another example.

Write the following in expanded form and evaluate the expression: \begin{align*}5^3\end{align*}53

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*}53=5×5×5

Then, multiply.

\begin{align*}5 \times 5 \times 5=125\end{align*}5×5×5=125

The answer is \begin{align*}5^3=5 \times 5 \times 5=125\end{align*}53=5×5×5=125.

Examples

Example 1

Earlier, you were given a problem about Sheila and her $3. 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*}$3×2.
  • After 2 months she will have \begin{align*}\$3 \times 2 \times 2\end{align*}$3×2×2 or \begin{align*}\$3 \times 2^2\end{align*}$3×22.
  • After 3 months she will have \begin{align*}\$3 \times 2 \times 2 \times 2\end{align*}$3×2×2×2 or \begin{align*}\$3 \times 2^3\end{align*}$3×23.

Continuing in this pattern you can see that

  • After 6 months she will have \begin{align*}\$3 \times 2^6\end{align*}$3×26.

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*}3×26=3×2×2×2×2×2×2

Now, multiply.

\begin{align*}3 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2=192\end{align*}3×2×2×2×2×2×2=192

The answer is that after 6 months of saving Sheila will have $192. Not bad!

Example 2

Evaluate: \begin{align*}2^3+4^2\end{align*}23+42

First, write the expression in expanded form.

\begin{align*}2^3+4^2=2 \times 2 \times 2+4 \times 4\end{align*}23+42=2×2×2+4×4

Next, multiply each part of the expression.

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

Then, add.

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

The answer is \begin{align*}2^3+4^2=24\end{align*}23+42=24.

Example 3

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

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*}35

The answer is \begin{align*}3 \times 3 \times 3 \times 3 \times 3=3^5\end{align*}3×3×3×3×3=35.

Example 4

Write the following in expanded form and evaluate the expression: \begin{align*}6^3\end{align*}63

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

\begin{align*}6^3=6 \times 6 \times 6\end{align*}63=6×6×6

Next, multiply.

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

The answer is \begin{align*}6^3=6 \times 6 \times 6=216\end{align*}63=6×6×6=216.

Example 5

Evaluate: \begin{align*}4^3-5^2\end{align*}4352

First, write the expression in expanded form.

\begin{align*}4^3-5^2=4 \times 4 \times 4-5 \times 5\end{align*}4352=4×4×45×5

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*}

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 1.7. 

Resources

Vocabulary

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.

Cubed

The cube of a number is the number multiplied by itself three times. For example, "two-cubed" = 2^3 = 2 \times 2 \times 2 = 8.

Power

The "power" refers to the value of the exponent. For example, 3^4 is "three to the fourth power".

Squared

Squared is the word used to refer to the exponent 2. For example, 5^2 could be read as "5 squared". When a number is squared, the number is multiplied by itself.

Whole Numbers

The whole numbers are all positive counting numbers and zero. The whole numbers are 0, 1, 2, 3, ...

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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