1. Operations with Real Numbers
1.1 Algebraic Expressions
Learning Objectives
By the end of this section it is expected that you will be able to:
- Use variables and algebraic symbols
- Identify expressions and equations
- Simplify expressions with exponents
- Simplify expressions using the order of operations
- Evaluate algebraic expressions
Use Variables and Algebraic Symbols
In algebra, letters of the alphabet are used to represent variables.
Letters often used for variables are .
Variables and Constants
A variable is a letter that represents a number or quantity whose value may change.
A constant is a number whose value always stays the same.
To write algebraically, we need some symbols as well as numbers and variables. There are several types of symbols we will be using. There are four basic arithmetic operations: addition, subtraction, multiplication, and division. We will summarize them here, along with words we use for the operations and the result.
| Operation | Notation | Say: | The result is… |
|---|---|---|---|
| Addition | the sum of and | ||
| Subtraction | the difference of and | ||
| Multiplication | The product of and | ||
| Division | divided by | The quotient of and |
In algebra, the cross symbol, ×, is not used to show multiplication because that symbol may cause confusion. Does mean (three times ) or (three times )? To make it clear, use • or parentheses for multiplication.When two quantities have the same value, we say they are equal and connect them with an equal sign.
Equality Symbol
The symbol is called the equal sign.
An inequality is used in algebra to compare two quantities that may have different values. The number line can help you understand inequalities. Remember that on the number line the numbers get larger as they go from left to right. So if we know that is greater than , it means that is to the right of on the number line. We use the symbols < and > for inequalities.
Inequality
< is read is less than
is to the left of on the number line
> is read is greater than
is to the right of on the number line
The expressions < > can be read from left-to-right or right-to-left, though in English we usually read from left-to-right. In general,
When we write an inequality symbol with a line under it, such as , it means or . We read this is less than or equal to . Also, if we put a slash through an equal sign, it means not equal.
We summarize the symbols of equality and inequality in the table below.
| Algebraic Notation | Say |
|---|---|
| is equal to | |
| is not equal to | |
| < | is less than |
| > | is greater than |
| is less than or equal to | |
| is greater than or equal to |
Symbols < and >
The symbols < and > each have a smaller side and a larger side.
smaller side < larger side
larger side > smaller side
The smaller side of the symbol faces the smaller number and the larger faces the larger number.
Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in written language. They indicate which expressions are to be kept together and separate from other expressions. The table below lists three of the most commonly used grouping symbols in algebra.
| Name | Symbol |
|---|---|
| parentheses | |
| brackets | |
| braces |
Here are some examples of expressions that include grouping symbols. We will simplify expressions like these later in this section.
Identify Expressions and Equations
What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. “Running very fast” is a phrase, but “The football player was running very fast” is a sentence. A sentence has a subject and a verb.
In algebra, we have expressions and equations. An expression is like a phrase. Here are some examples of expressions and how they relate to word phrases:
| Expression | Words | Phrase |
|---|---|---|
| the sum of three and five | ||
| minus one | the difference of and one | |
| the product of six and seven | ||
| divided by | the quotient of and |
Notice that the phrases do not form a complete sentence because the phrase does not have a verb. An equation is two expressions linked with an equal sign. When you read the words the symbols represent in an equation, you have a complete sentence in English. The equal sign gives the verb. Here are some examples of equations:
| Equation | Sentence |
|---|---|
| The sum of three and five is equal to eight. | |
| minus one equals fourteen. | |
| The product of six and seven is equal to forty-two. | |
| is equal to fifty-three. | |
| plus nine is equal to two minus three. |
Expressions and Equations
An expression is a number, a variable, or a combination of numbers and variables and operation symbols.
An equation is made up of two expressions connected by an equal sign.
EXAMPLE 1
Determine if each is an expression or an equation:
| a. | This is an equation—two expressions are connected with an equal sign. |
| b. | This is an expression—no equal sign. |
| c. | This is an expression—no equal sign. |
| d. | This is an equation—two expressions are connected with an equal sign. |
TRY IT 1
Determine if each is an expression or an equation:
Show Answer
- equation
- expression
Simplify Expressions with Exponents
To simplify a numerical expression means to do all the math possible. For example, to simplify we’d first multiply to get and then add the to get . A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this:
Suppose we have the expression . We could write this more compactly using exponential notation. Exponential notation is also called power and is used in algebra to represent a quantity multiplied by itself several times. We write as and as . In expressions such as , the is called the base and the is called the exponent. The exponent tells us how many factors of the base we have to multiply.
We say is in exponential notation and is in expanded notation.
Exponential Notation (Power)
For any expression is a factor multiplied by itself times if is a positive integer.
The expression is read to the power.
For powers of and , we have special names.
The table below lists some examples of expressions written in exponential notation.
| Exponential Notation | In Words |
|---|---|
| to the second power, or squared | |
| to the third power, or cubed | |
| to the fourth power | |
| to the fifth power |
EXAMPLE 2
Write each expression in exponential form:
| a. The base 16 is a factor 7 times. | |
| b. The base 9 is a factor 5 times. | |
| c. The base is a factor 4 times. | |
| d. The base is a factor 8 times. |
TRY IT 2
Write each expression in exponential form:
Show Answer
415
EXAMPLE 3
Write each exponential expression in expanded form:
a. The base is and the exponent is , so means
b. The base is and the exponent is , so means
TRY IT 3
Write each exponential expression in expanded form:
Show Answer
To simplify an exponential expression without using a calculator, we write it in expanded form and then multiply the factors.
EXAMPLE 4
Simplify: .
| Expand the expression. | |
| Multiply left to right. | |
| Multiply. |
TRY IT 4
Simplify:
Show Answer
- 125
- 1
Simplify Expressions Using the Order of Operations
We’ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values.
For example, consider the expression:
Imagine the confusion that could result if every problem had several different correct answers. The same expression should give the same result. So mathematicians established some guidelines called the order of operations, which outlines the order in which parts of an expression must be simplified.
Order of Operations
When simplifying mathematical expressions perform the operations in the following order:
1. Parentheses and other Grouping Symbols
- Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
2. Exponents
- Simplify all expressions with exponents.
3. Multiplication and Division
- Perform all multiplication and division in order from left to right. These operations have equal priority.
4. Addition and Subtraction
- Perform all addition and subtraction in order from left to right. These operations have equal priority.
Students often ask, “How will I remember the order?” Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase.
Please Excuse My Dear Aunt Sally.
| Please | Parentheses |
| Excuse | Exponents |
| My Dear | Multiplication and Division |
| Aunt Sally | Addition and Subtraction |
It’s good that ‘My Dear’ goes together, as this reminds us that multiplication and division have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.
Similarly, ‘Aunt Sally’ goes together and so reminds us that addition and subtraction also have equal priority and we do them in order from left to right.
EXAMPLE 5
Simplify the expressions:
| a. | |
| Are there any parentheses? No. | |
| Are there any exponents? No. | |
| Is there any multiplication or division? Yes. | |
| Multiply first. | |
| Add. | |
| b. | |
| Are there any parentheses? Yes. | |
| Simplify inside the parentheses. | |
| Are there any exponents? No. | |
| Is there any multiplication or division? Yes. | |
| Multiply. |
TRY IT 5
Simplify the expressions:
Show Answer
- 2
- 14
EXAMPLE 6
Simplify: .
| Parentheses? Yes, subtract first. | |
| Exponents? No. | |
| Multiplication or division? Yes. | |
| Divide first because we multiply and divide left to right. | |
| Any other multiplication or division? Yes. | |
| Multiply. | |
| Any other multiplication or division? No. | |
| Any addition or subtraction? Yes. |
TRY IT 6
Simplify:
Show Answer
16
When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.
EXAMPLE 7
.
| Are there any parentheses (or other grouping symbol)? Yes. | |
| Focus on the parentheses that are inside the brackets. | |
| Subtract. | |
| Continue inside the brackets and multiply. | |
| Continue inside the brackets and subtract. | |
| The expression inside the brackets requires no further simplification. | |
| Are there any exponents? Yes. | |
| Simplify exponents. | |
| Is there any multiplication or division? Yes. | |
| Multiply. | |
| Is there any addition or subtraction? Yes. | |
| Add. | |
| Add. | |
TRY IT 7
Simplify:
Show Answer
86
EXAMPLE 8
Simplify: .
| If an expression has several exponents, they may be simplified in the same step. | |
| Simplify exponents. | |
| Divide. | |
| Add. | |
| Subtract. | |
TRY IT 8
Simplify:
Show Answer
81
Evaluate Algebraic Expressions
In the last section, we simplified expressions using the order of operations. In this section, we’ll evaluate expressions—again following the order of operations.
To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number. To evaluate an expression, we substitute the given number for the variable in the expression and then simplify the expression using the order of operations.
EXAMPLE 9
Evaluate
Remember means times , so means times .
a. To evaluate the expression when , we substitute for , and then simplify.
| Multiply. | |
| Subtract. |
b. To evaluate the expression when , we substitute for , and then simplify.
| Multiply. | |
| Subtract. |
Notice that in part a) that we wrote and in part b) we wrote . Both the dot and the parentheses tell us to multiply.
TRY IT 9
Evaluate:
Show Answer
- 13
- 5
EXAMPLE 10
Evaluate when .
We substitute for , and then simplify the expression.
| Use the definition of exponent. | |
| Multiply. |
When , the expression has a value of .
TRY IT 10
Evaluate:
.
Show Answer
64
EXAMPLE 11
.
In this expression, the variable is an exponent.
| Use the definition of exponent. | |
| Multiply. |
When , the expression has a value of .
TRY IT 11
Evaluate:
.
Show Answer
64
EXAMPLE 12
.
This expression contains two variables, so we must make two substitutions.
| Multiply. | |
| Add and subtract left to right. |
When and , the expression has a value of .
TRY IT 12
Evaluate:
Show Answer
33
EXAMPLE 13
.
We need to be careful when an expression has a variable with an exponent. In this expression, means and is different from the expression , which means .
| Simplify . | |
| Multiply. | |
| Add. |
TRY IT 13
Evaluate:
.
Show Answer
40
ACCESS ADDITIONAL ONLINE RESOURCES
Key Concepts
| Operation | Notation | Say: | The result is… |
|---|---|---|---|
| Addition | the sum of and | ||
| Multiplication | The product of and | ||
| Subtraction | the difference of and | ||
| Division | divided by | The quotient of and |
- Equality Symbol
- is read as is equal to
- The symbol is called the equal sign.
- Inequality
- < is read is less than
- is to the left of on the number line
- > is read is greater than
- is to the right of on the number line
| Algebraic Notation | Say |
|---|---|
| is equal to | |
| is not equal to | |
| < | is less than |
| > | is greater than |
| is less than or equal to | |
| is greater than or equal to |
- Exponential Notation
- For any expression is a factor multiplied by itself times, if is a positive integer.
- means multiply factors of
- The expression of is read to the power.
Order of Operations When simplifying mathematical expressions perform the operations in the following order:
- Parentheses and other Grouping Symbols: Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
- Exponents: Simplify all expressions with exponents.
- Multiplication and Division: Perform all multiplication and division in order from left to right. These operations have equal priority.
- Addition and Subtraction: Perform all addition and subtraction in order from left to right. These operations have equal priority.
Glossary
- expressions
- An expression is a number, a variable, or a combination of numbers and variables and operation symbols.
- equation
- An equation is made up of two expressions connected by an equal sign.
1.1 Exercise Set
In the following exercises, determine if each is an expression or an equation.
In the following exercises, write in exponential form.
In the following exercises, write in expanded form.
In the following exercises, simplify.
In the following exercises, evaluate the expression for the given value.
Answers:
- equation
- expression
- expression
- equation
- 37
- x5
- 5 × 5 × 5
- 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
-
- 43
- 55
- 5
- 34
- 58
- 6
- 13
- 4
- 35
- 10
- 41
- 81
- 149
- 50
- 22
- 26
- 144
- 27
- 21
- 41
- 9
- 73
- 54
Attributions
This chapter has been adapted from “Use the Language of Algebra” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.