11.1.1: Exponential Notation

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Learning Objectives

Evaluate expressions containing exponents.
Evaluate exponential notations with exponents 0 of 1 and .
Write an exponential expression involving negative exponents with positive exponents.

Introduction

A common language is needed in order to communicate mathematical ideas clearly and efficiently. Exponential notation is one example. It was developed to write repeated multiplication more efficiently. For example, growth occurs in living organisms by the division of cells. One type of cell divides 2 times in an hour. So in 12 hours, the cell will divide $\ 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$ times. This can be written more efficiently as $\ 2^{12}$.

Exponential Vocabulary

We use exponential notation to write repeated multiplication, such as $\ 10 \cdot 10 \cdot 10$ as $\ 10^{3}$. The 10 in $\ 10^{3}$ is called the base. The 3 in $\ 10^{3}$ is called the exponent. The expression $\ 10^{3}$ is called the exponential expression.

$Screen Shot 2021-06-06 at 2.31.40 PM.png$

$\ 10^{3}$ is read as “10 to the third power” or “10 raised to the power of 3” or “10 cubed.” It means $\ 10 \cdot 10 \cdot 10$, or 1,000.
$\ 8^{2}$ is read as “8 to the second power” or “8 raised to the power of 2” or “8 squared.” It means $\ 8 \cdot 8$, or 64.
$\ 5^{4}$ is read as “5 to the fourth power” or “5 raised to the power of 4.” It means $\ 5 \cdot 5 \cdot 5 \cdot 5$, or 625.
$\ b^{5}$ is read as “$\ b$ to the fifth power” or “$\ b$ raised to the power of 5.” It means $\ b \cdot b \cdot b \cdot b \cdot b$. Its value will depend on the value of $\ b$.

The exponent applies only to the number that it is next to. So in the expression $\ x y^{4}$, only the $\ y$ is affected by the 4. $\ x y^{4}$ means $\ x \cdot y \cdot y \cdot y \cdot y$.

If the exponential expression is negative, such as $\ -3^{4}$, it means $\ -(3 \cdot 3 \cdot 3 \cdot 3)$ or -81.

If -3 is to be the base of the exponential expression, it must be written as $\ (-3)^{4}$, which means $\ -3 \cdot-3 \cdot-3 \cdot-3$, or 81.

Likewise, $\ (-x)^{4}=(-x) \cdot(-x) \cdot(-x) \cdot(-x)=x^{4}$. while $\ -x^{4}=-(x \cdot x \cdot x \cdot x)$.

You can see that there is quite a difference, so you have to be very careful!

Evaluating Expressions Containing Exponents

Evaluating expressions containing exponents is the same as evaluating any expression. You substitute the value of the variable into the expression and simplify.

You can use PEMDAS to remember the order in which you should evaluate the expression. First, evaluate anything in Parentheses or grouping symbols. Next, look for Exponents, followed by Multiplication and Division (reading from left to right), and lastly, Addition and Subtraction (again, reading from left to right).

So, when you evaluate the expression $\ 5 x^{3}$ if $\ x=4$, first substitute the value 4 for the variable $\ x$. Then evaluate, using order of operations.

Example

Evaluate. $\ 5 x^{3} \text { if } x=4$

Solution

$\ 5 \cdot 4^{3}$	Substitute 4 for the variable $\ x$.
$\ 5(4 \cdot 4 \cdot 4)=5 \cdot 64$	Evaluate $\ 4^{3}$.
$\ 320$	Multiply.

$\ 5 x^{3}=320 \text { when } x=4$

Notice the difference between the example above and the one below.

Example

Evaluate. $\ (5 x)^{3} \text { if } x=4$

Solution

$\ (5 \cdot 4)^{3}$	Substitute 4 for the variable $\ x$.
$\ 20^{3}$	Multiply.
$\ 20 \cdot 20 \cdot 20=8,000$	Evaluate $\ 20^{3}$.

$\ (5 x)^{3}=8,000 \text { when } x=4$

The addition of parentheses made quite a difference!

Example

Evaluate. $\ x^{3} \text { if } x=-4$

Solution

$\ (-4)^{3}$	Substitute -4 for the variable $\ x$.
$\ -4 \cdot-4 \cdot-4$	Evaluate.
$\ -4 \cdot-4 \cdot-4=-64$	Multiply.

$\ x^{3}=-64, \text { when } x=-4$

Exercise

Evaluate the expression $\ -(2 x)^{4}, \text { if } x=3$.

1,296
-1,296
162
-162

Answer

Incorrect. Substitute the value of 3 for the variable $\ x$ and evaluate $\ -(2 \cdot 3)^{4}$. Do not apply the negative sign until after you have evaluated the expression $\ (6)^{4}$. The correct answer is -1,296.
Correct. Substitute the value of 3 for the variable $\ x$ and evaluate $\ -(2 \cdot 3)^{4}=-6^{4}=-1,296$.
Incorrect. Substitute the value of 3 for the variable $\ x$ and evaluate $\ -(2 \cdot 3)^{4}$. Apply the exponent 4 to the product $\ 2 \cdot 3$, or 6. Then apply the negative sign. The correct answer is -1,296.
Incorrect. Substitute the value of 3 for the variable $\ x$ and evaluate $\ -(2 \cdot 3)^{4}$. Apply the exponent 4 to the product $\ 2 \cdot 3$, or 6. Then apply the negative sign. The correct answer is -1,296.

Exponents of Zero and One

What does it mean when an exponent is 0 or 1? Let’s consider $\ 25^{1}$. Any value raised to the power of 1 is just the value itself. This makes sense, because the exponent of 1 means the base is used as a factor only once. So the base stands alone, and $\ 25^{1}$ is simply 25.

But what about a value raised to the power of 0? Use what you know about powers of 10 to find out what the power of 0 means. Below is a list of powers of 10 and their equivalent values. Look at how the numbers change going down the left and right columns. Can you notice a pattern there?

$\ \begin{array}{ccr}
\bf\text { Exponential Form } & \bf\text { Expanded Form } & \bf\text { Value } \\
10^{5} & 10 \cdot 10 \cdot 10 \cdot 10 \cdot 10 & 100,000 \\
10^{4} & 10 \cdot 10 \cdot 10 \cdot 10 & 10,000 \\
10^{3} & 10 \cdot 10 \cdot 10 & 1,000 \\
10^{2} & 10 \cdot 10 & 100 \\
10^{1} & 10 & 10
\end{array}$

Moving down the table, each row drops one factor of 10 from the one above it. From row 1 to row 2, the exponential form goes from $\ 10^{5}$ to $\ 10^{4}$. The value drops from 100,000 to 10,000. Another way to put this is that each value is divided by 10 to produce the next value down the column.

Let’s use this pattern of division by 10 to predict the value of $\ 10^{0}$.

Following the pattern, you see that $\ 10^{0}$ is equal to 1. Would the pattern hold for a different base? Say a base of 3?

$\ \begin{array}{ccr}
\bf\text { Exponential Form } & \bf\text { Expanded Form } & \bf\text { Value } \\
3^{5} & 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 & 243 \\
3^{4} & 3 \cdot 3 \cdot 3 \cdot 3 & 81 \\
3^{3} & 3 \cdot 3 \cdot 3 & 27 \\
3^{2} & 3 \cdot 3 & 9 \\
3^{1} & 3 & 3 \\
\hline
3^{0} & 1 & 1 \\
\hline
\end{array}$

Yes! And the same pattern would hold true for any non-zero number or variable raised to a power of 0, $\ n^{0}=1$.

There is a conflict when the base is 0. You know that $\ 0^{3}=0$, $\ 0^{2}=0$, and $\ 0^{1}=0$, so you would expect $\ 0^{0}$ to also be equal to 0. However, the above pattern says that any base raised to the power of 0 is 1, so this leads you to believe that $\ 0^{0}=1$. Notice the competing patterns: $\ 0^{0}$ cannot be both 0 and 1! In this case, mathematicians say that the value of $\ 0^{0}$ is undefined. (And remember that undefined is not the same as 0!)

Exponents of 0 or 1

Any number or variable raised to a power of 1 is the number itself. $\ n^{1}=n$

Any non-zero number or variable raised to a power of 0 is equal to 1. $\ n^{0}=1$

The quantity $\ 0^{0}$ is undefined.

Example

Evaluate. $\ 2 x^{0} \text { if } x=9$

Solution

$\ 2 \cdot 9^{0}$	Substitute 9 for the variable $\ x$.
$\ 2 \cdot 1$	Evaluate $\ 9^{0}$.
$\ 2$	Multiply.

$\ 2 x^{0}=2, \text { if } x=9$

As done previously, to evaluate expressions containing exponents of 0 or 1, substitute the value of the variable into the expression and simplify.

Exercise

Evaluate the expression $\ 3 x^{0}-y^{1}, \text { if } x=12 \text { and } y=-6$.

42
-3
9
2

Answer

Incorrect. Substitute the value of 12 for the variable $\ x$ and -6 for the variable $\ y$: $\ 3 \cdot 12^{0}-(-6)^{1}$. Remember that $\ 12^{0}=1$. The correct answer is 9.
Incorrect. Substitute the value of 12 for the variable $\ x$ and -6 for the variable $\ y$: $\ 3 \cdot 12^{0}-(-6)^{1}$. Remember that $\ (-6)^{1}=-6$ and $\ -(-6)=6$. The correct answer is 9.
Correct. $\ 3 \cdot 12^{0}-(-6)^{1}=3 \cdot 1-(-6)^{1}=3 \cdot 1+6=9$
Incorrect. Substitute the value of 12 for the variable $\ x$ and -6 for the variable $\ y$: $\ 3 \cdot 12^{0}-(-6)^{1}$. Remember that a base raised to the power of 1 is the base. The correct answer is 9.

Negative Exponents

What does it mean when an exponent is a negative integer? Let’s use the powers of 10 pattern from earlier to find out. If you continue this pattern to add some more rows, beyond $\ 10^{0}$, you find the following:

$\ \begin{array}{ccr}
\bf\text{Exponential Form}&\bf\text{Expanded Form}&\bf\text{Value}\\
10^{5} & 10 \cdot 10 \cdot 10 \cdot 10 \cdot 10 & 100,000 \\
10^{4} & 10 \cdot 10 \cdot 10 \cdot 10 & 10,000 \\
10^{3} & 10 \cdot 10 \cdot 10 & 1,000 \\
10^{2} & 10 \cdot 10 & 100 \\
10^{1} & 10 & 10 \\
10^{0} & 1 & 1 \\
10^{-1} & \frac{1}{10^{1}} & \frac{1}{10} \\
10^{-2} & \frac{1}{10^{2}} & \frac{1}{100}
\end{array}$

Following the pattern, you see that $\ 10^{0}$ is equal to 1. Then you get into negative exponents: $\ 10^{-1}$ is equal to $\ \frac{1}{10^{1}}$, and $\ 10^{-2}$ is the same as (\ \frac{1}{10^{2}}\).

Following this pattern, a number with a negative exponent can be rewritten as the reciprocal of the original number, with a positive exponent.

For example, $\ 10^{-3}=\frac{1}{10^{3}}$ and $\ 10^{-7}=\frac{1}{10^{7}}$.

To see if these patterns hold true for numbers other than 10, check out this table with powers of 3.

The numbers are different but the patterns are the same. We are now ready to state the definition of a negative exponent.

Negative Exponent

For any non-zero number $\ n$ and any integer $\ x$, $\ n^{-x}=\frac{1}{n^{x}}$. For example, $\ 5^{-2}=\frac{1}{5^{2}}$.

Note that the definition above states that the base, $\ n$ must be a “non-zero number.”

Exercise

Evaluate the expression $\ \left(x^{-2}\right) \cdot\left(x^{0}\right)$ when $\ x=6$.

$\ \frac{1}{36}$
$\ \frac{1}{6}$
$\ 0$
$\ 36$

Answer

Correct. Substitute the value of 6 for the variable $\ x$ and evaluate. $\ 6^{-2} \cdot 6^{0}=\frac{1}{6^{2}} \cdot 6^{0}=\frac{1}{36} \cdot 1=\frac{1}{36}$.
Incorrect. Substitute the value of 6 for the variable $\ x$ and evaluate: $\ 6^{-2} \cdot 6^{0}$. Remember that $\ (6)^{0}=1$. The correct answer is $\ \frac{1}{36}$.
Incorrect. Substitute the value of 6 for the variable $\ x$ and evaluate: $\ 6^{-2} \cdot 6^{0}$. Remember that $\ (6)^{0}=1$. The correct answer is $\ \frac{1}{36}$.
Incorrect. Substitute the value of 6 for the variable $\ x$ and then evaluate: $\ 6^{-2} \cdot 6^{0}$ Remember that $\ (6)^{-2}=\frac{1}{6^{2}}=\frac{1}{36}$. The correct answer is $\ \frac{1}{36}$.

Summary

Exponential notation is composed of a base and an exponent. It is a “shorthand” way of writing repeated multiplication, and indicates that the base is a factor and the exponent is the number of times the factor is used in the multiplication. The basic rules of exponents are as follows:

An exponent applies only to the value to its immediate left.
When a quantity in parentheses is raised to a power, the exponent applies to everything inside the parentheses.
For any non-zero number $\ n$, $\ n^{0}=1$.
For any non-zero number $\ n$ and any integer $\ x$, $\ n^{-x}=\frac{1}{n^{x}}$.

\(\ 5 \cdot 4^{3}\)	Substitute 4 for the variable \(\ x\).
\(\ 5(4 \cdot 4 \cdot 4)=5 \cdot 64\)	Evaluate \(\ 4^{3}\).
\(\ 320\)	Multiply.

\(\ (5 \cdot 4)^{3}\)	Substitute 4 for the variable \(\ x\).
\(\ 20^{3}\)	Multiply.
\(\ 20 \cdot 20 \cdot 20=8,000\)	Evaluate \(\ 20^{3}\).

\(\ (-4)^{3}\)	Substitute -4 for the variable \(\ x\).
\(\ -4 \cdot-4 \cdot-4\)	Evaluate.
\(\ -4 \cdot-4 \cdot-4=-64\)	Multiply.

\(\ 2 \cdot 9^{0}\)	Substitute 9 for the variable \(\ x\).
\(\ 2 \cdot 1\)	Evaluate \(\ 9^{0}\).
\(\ 2\)	Multiply.

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