1.1 Algebraic Expressions

Izabela Mazur; Kim Moshenko

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 $x,y,a,b,\text{and}\phantom{\rule{0.2em}{0ex}}c$ .

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	$a+b$	$a\phantom{\rule{0.2em}{0ex}}\text{plus}\phantom{\rule{0.2em}{0ex}}b$	the sum of $a$ and $b$
Subtraction	$a-b$	$a\phantom{\rule{0.2em}{0ex}}\text{minus}\phantom{\rule{0.2em}{0ex}}b$	the difference of $a$ and $b$
Multiplication	$a\cdot b,\left(a\right)\left(b\right),\left(a\right)b,a\left(b\right)$	$a\phantom{\rule{0.2em}{0ex}}\text{times}\phantom{\rule{0.2em}{0ex}}b$	The product of $a$ and $b$
Division	$a\div b,a/b,\phantom{\rule{0.2em}{0ex}}\frac{a}{b},b\overline{)a}$	$a$ divided by $b$	The quotient of $a$ and $b$

In algebra, the cross symbol, ×, is not used to show multiplication because that symbol may cause confusion. Does $3xy$ mean $3 \phantom{\rule{0.2em}{0ex}} \times \phantom{\rule{0.2em}{0ex}}y$ (three times $y$ ) or $3 \cdot x \cdot y$ (three times $x\phantom{\rule{0.2em}{0ex}}\text{times}\phantom{\rule{0.2em}{0ex}}y$ )? 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

$a=b\phantom{\rule{0.2em}{0ex}}\text{is read}\phantom{\rule{0.2em}{0ex}}a\phantom{\rule{0.2em}{0ex}}\text{is equal to}\phantom{\rule{0.2em}{0ex}}b$

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 $b$ is greater than $a$ , it means that $b$ is to the right of $a$ on the number line. We use the symbols < and > for inequalities.

Inequality

$a$ < $b$ is read $a$ is less than $b$

$a$ is to the left of $b$ on the number line

The figure shows a horizontal number line that begins with the letter a on the left then the letter b to its right.

$a$ > $b$ is read $a$ is greater than $b$

$a$ is to the right of $b$ on the number line

The figure shows a horizontal number line that begins with the letter b on the left then the letter a to its right.

The expressions $a$ < $b\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}a$ > $\phantom{\rule{0.2em}{0ex}}b$ can be read from left-to-right or right-to-left, though in English we usually read from left-to-right. In general,

"7 is less than 11" equivalent to "11 is greater than 7"

When we write an inequality symbol with a line under it, such as $a\le b$ , it means $a<b$ or $a=b$ . We read this $a$ is less than or equal to $b$ . Also, if we put a slash through an equal sign, $\ne$ it means not equal.

We summarize the symbols of equality and inequality in the table below.

Algebraic Notation

Say

$a=b$

$a$ is equal to $b$

$a\ne b$

$a$ is not equal to $b$

$a$ < $b$

$a$ is less than $b$

$a$ > $b$

$a$ is greater than $b$

$a\le b$

$a$ is less than or equal to $b$

$a\ge b$

$a$ is greater than or equal to $b$

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.

Common Grouping Symbols
Name	Symbol
parentheses	$\left(\phantom{\rule{0.5em}{0ex}}\right)$
brackets	$\left[\phantom{\rule{0.5em}{0ex}}\right]$
braces	$\left\{\phantom{\rule{0.5em}{0ex}}\right\}$

Here are some examples of expressions that include grouping symbols. We will simplify expressions like these later in this section.

$8\left(14-8\right)\phantom{\rule{4em}{0ex}}21-3\left[2+4\left(9-8\right)\right]\phantom{\rule{4em}{0ex}}24\div \left\{13-2\left[1\left(6-5\right)+4\right]\right\}$

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
$3+5$	$3\phantom{\rule{0.2em}{0ex}}\text{plus}\phantom{\rule{0.2em}{0ex}}5$	the sum of three and five
$n-1$	$n$ minus one	the difference of $n$ and one
$6\cdot 7$	$6\phantom{\rule{0.2em}{0ex}}\text{times}\phantom{\rule{0.2em}{0ex}}7$	the product of six and seven
$\frac{x}{y}$	$x$ divided by $y$	the quotient of $x$ and $y$

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
$3+5=8$	The sum of three and five is equal to eight.
$n-1=14$	$n$ minus one equals fourteen.
$6\cdot 7=42$	The product of six and seven is equal to forty-two.
$x=53$	$x$ is equal to fifty-three.
$y+9=2y-3$	$y$ plus nine is equal to two $y$ 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:

$\phantom{\rule{0.2em}{0ex}}16-6=10$
$\phantom{\rule{0.2em}{0ex}}4\cdot 2+1$
$\phantom{\rule{0.2em}{0ex}}x\div 25$
$\phantom{\rule{0.2em}{0ex}}y+8=40$

Solution

a. $\phantom{\rule{0.2em}{0ex}}16-6=10$	This is an equation—two expressions are connected with an equal sign.
b. $\phantom{\rule{0.2em}{0ex}}4\cdot 2+1$	This is an expression—no equal sign.
c. $\phantom{\rule{0.2em}{0ex}}x\div 25$	This is an expression—no equal sign.
d. $\phantom{\rule{0.2em}{0ex}}y+8=40$	This is an equation—two expressions are connected with an equal sign.

TRY IT 1

Determine if each is an expression or an equation:

$\phantom{\rule{0.2em}{0ex}}23+6=29\phantom{\rule{0.4em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}7\cdot 3-7$

Show Answer

equation
expression

Simplify Expressions with Exponents

To simplify a numerical expression means to do all the math possible. For example, to simplify $4\cdot2+1$ we’d first multiply $4\cdot2$ to get $8$ and then add the $1$ to get $9$ . 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:

$4\cdot2+1$

$8+1$

$9$

Suppose we have the expression $2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2$ . 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 $2\cdot2\cdot2$ as ${2}^{3}$ and $2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2$ as ${2}^{9}$ . In expressions such as ${2}^{3}$ , the $2$ is called the base and the $3$ is called the exponent. The exponent tells us how many factors of the base we have to multiply.

The image shows the number two with the number three, in superscript, to the right of the two. The number two is labeled as “base” and the number three is labeled as “exponent”.

$\text{means multiply three factors of 2}$

We say ${2}^{3}$ is in exponential notation and $2\cdot2\cdot2$ is in expanded notation.

Exponential Notation (Power)

For any expression ${a}^{n},a$ is a factor multiplied by itself $n$ times if $n$ is a positive integer.

${a}^{n}\text{means multiply}\phantom{\rule{0.2em}{0ex}}n\phantom{\rule{0.2em}{0ex}}\text{factors of}\phantom{\rule{0.2em}{0ex}}a$

At the top of the image is the letter a with the letter n, in superscript, to the right of the a. The letter a is labeled as “base” and the letter n is labeled as “exponent”. Below this is the letter a with the letter n, in superscript, to the right of the a set equal to n factors of a.

The expression ${a}^{n}$ is read $a$ to the ${n}^{th}$ power.

For powers of $n=2$ and $n=3$ , we have special names.

$\begin{array}{l}{a}^{2}\phantom{\rule{0.2em}{0ex}}\text{is read as}\phantom{\rule{0.2em}{0ex}}\text{"}a\phantom{\rule{0.2em}{0ex}}\text{squared"}\\ {a}^{3}\phantom{\rule{0.2em}{0ex}}\text{is read as}\phantom{\rule{0.2em}{0ex}}\text{"}a\phantom{\rule{0.2em}{0ex}}\text{cubed"}\end{array}$

The table below lists some examples of expressions written in exponential notation.

Exponential Notation	In Words
${7}^{2}$	$7$ to the second power, or $7$ squared
${5}^{3}$	$5$ to the third power, or $5$ cubed
${9}^{4}$	$9$ to the fourth power
${12}^{5}$	$12$ to the fifth power

EXAMPLE 2

Write each expression in exponential form:

$\phantom{\rule{0.2em}{0ex}}16\cdot16\cdot16\cdot16\cdot16\cdot16\cdot16$
$\phantom{\rule{0.2em}{0ex}}9\cdot9\cdot9\cdot9\cdot9$
$\phantom{\rule{0.2em}{0ex}}x\cdot x\cdot x\cdot x$
$\phantom{\rule{0.2em}{0ex}}a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a$

Solution

a. The base 16 is a factor 7 times.	${16}^{7}$
b. The base 9 is a factor 5 times.	${9}^{5}$
c. The base $x$ is a factor 4 times.	${x}^{4}$
d. The base $a$ is a factor 8 times.	${a}^{8}$

TRY IT 2

Write each expression in exponential form:

$41\cdot41\cdot41\cdot41\cdot41$

Show Answer

41⁵

EXAMPLE 3

Write each exponential expression in expanded form:

$\phantom{\rule{0.2em}{0ex}}{8}^{6}\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}{x}^{5}$

Solution

a. The base is $8$ and the exponent is $6$ , so ${8}^{6}$ means $8\cdot 8\cdot 8\cdot 8\cdot 8\cdot 8$

b. The base is $x$ and the exponent is $5$ , so ${x}^{5}$ means $x\cdot x\cdot x\cdot x\cdot x$

TRY IT 3

Write each exponential expression in expanded form:

$\phantom{\rule{0.2em}{0ex}}{4}^{8}$
$\phantom{\rule{0.2em}{0ex}}{a}^{7}$

Show Answer

$4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4$
$a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a$

To simplify an exponential expression without using a calculator, we write it in expanded form and then multiply the factors.

EXAMPLE 4

Simplify: ${3}^{4}$ .

Solution

	${3}^{4}$
Expand the expression.	$3\cdot 3\cdot 3\cdot 3$
Multiply left to right.	$9\cdot 3\cdot 3$
	$27\cdot 3$
Multiply.	$81$

TRY IT 4

Simplify:

$\phantom{\rule{0.2em}{0ex}}{5}^{3}\phantom{\rule{0.4em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}{1}^{7}$

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:

$4+3\cdot 7$

$\begin{array}{cccc}\hfill \text{Some students say it simplifies to 49.}\hfill & \phantom{\rule{2em}{0ex}}& & \hfill \text{Some students say it simplifies to 25.}\hfill \\ \begin{array}{ccc}& & \hfill 4+3\cdot 7\hfill \\ \text{Since}\phantom{\rule{0.2em}{0ex}}4+3\phantom{\rule{0.2em}{0ex}}\text{gives 7.}\hfill & \phantom{\rule{2em}{0ex}}& \hfill 7\cdot 7\hfill \\ \text{And}\phantom{\rule{0.2em}{0ex}}7\cdot 7\phantom{\rule{0.2em}{0ex}}\text{is 49.}\hfill & \phantom{\rule{2em}{0ex}}& \hfill 49\hfill \end{array}& & & \begin{array}{ccc}& & \hfill 4+3\cdot 7\hfill \\ \phantom{\rule{0.2em}{0ex}}\text{Since}\phantom{\rule{0.2em}{0ex}}3\cdot 7\phantom{\rule{0.2em}{0ex}}\text{is 21.}\hfill & & \hfill 4+21\hfill \\ \phantom{\rule{0.2em}{0ex}}\text{And}\phantom{\rule{0.2em}{0ex}}21+4\phantom{\rule{0.2em}{0ex}}\text{makes 25.}\hfill & & \hfill 25\hfill \end{array}\hfill \end{array}$

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:

$\phantom{\rule{0.2em}{0ex}}4+3\cdot 7\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}\left(4+3\right)\cdot 7$

Solution

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:

$\phantom{\rule{0.2em}{0ex}}12-5\cdot 2\phantom{\rule{0.4em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}\left(12-5\right)\cdot 2$

Show Answer

2
14

EXAMPLE 6

Simplify: $18\div 6+4\left(5-2\right)$ .

Solution


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:

$30\div 5+10\left(3-2\right)$

Show Answer

16

When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.

EXAMPLE 7

$\text{Simplify:}\phantom{\rule{0.2em}{0ex}}5+{2}^{3}+3\left[6-3\left(4-2\right)\right]$ .

Solution


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:

$9+{5}^{3}-\left[4\left(9+3\right)\right]$

Show Answer

86

EXAMPLE 8

Simplify: ${2}^{3}+{3}^{4}\div 3-{5}^{2}$ .

Solution


If an expression has several exponents, they may be simplified in the same step.
Simplify exponents.
Divide.
Add.
Subtract.

TRY IT 8

Simplify:

${3}^{2}+{2}^{4}\div 2+{4}^{3}$

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 $9x-2,\text{when}\phantom{\rule{0.2em}{0ex}}$

$\phantom{\rule{0.2em}{0ex}}x=5\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}x=1$

Solution

Remember $ab$ means $a$ times $b$ , so $9x$ means $9$ times $x$ .

a. To evaluate the expression when $x=5$ , we substitute $5$ for $x$ , and then simplify.



Multiply.
Subtract.

b. To evaluate the expression when $x=1$ , we substitute $1$ for $x$ , and then simplify.



Multiply.
Subtract.

Notice that in part a) that we wrote $9\cdot 5$ and in part b) we wrote $9\left(1\right)$ . Both the dot and the parentheses tell us to multiply.

TRY IT 9

Evaluate:

$8x-3,\text{when}\phantom{\rule{0.2em}{0ex}}$

$\phantom{\rule{0.2em}{0ex}}x=2\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}x=1$

Show Answer

13
5

EXAMPLE 10

Evaluate ${x}^{2}$ when $x=10$ .

Solution

We substitute $10$ for $x$ , and then simplify the expression.



Use the definition of exponent.
Multiply.

When $x=10$ , the expression ${x}^{2}$ has a value of $100$ .

TRY IT 10

Evaluate:

${x}^{2}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=8$ .

Show Answer

64

EXAMPLE 11

$\text{Evaluate}\phantom{\rule{0.2em}{0ex}}{2}^{x}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=5$ .

Solution

In this expression, the variable is an exponent.



Use the definition of exponent.
Multiply.

When $x=5$ , the expression ${2}^{x}$ has a value of $32$ .

TRY IT 11

Evaluate:

${2}^{x}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=6$ .

Show Answer

64

EXAMPLE 12

$\text{Evaluate}\phantom{\rule{0.2em}{0ex}}3x+4y-6\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=10\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=2$ .

Solution

This expression contains two variables, so we must make two substitutions.



Multiply.
Add and subtract left to right.

When $x=10$ and $y=2$ , the expression $3x+4y-6$ has a value of $32$ .

TRY IT 12

Evaluate:

$2x+5y-4\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=11\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=3$

Show Answer

33

EXAMPLE 13

$\text{Evaluate}\phantom{\rule{0.2em}{0ex}}2{x}^{2}+3x+8\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=4$ .

Solution

We need to be careful when an expression has a variable with an exponent. In this expression, $2{x}^{2}$ means $2\cdot x\cdot x$ and is different from the expression ${\left(2x\right)}^{2}$ , which means $2x\cdot 2x$ .



Simplify ${4}^{2}$ .
Multiply.
Add.

TRY IT 13

Evaluate:

$3{x}^{2}+4x+1\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=3$ .

Show Answer

40

ACCESS ADDITIONAL ONLINE RESOURCES

Key Concepts

Operation

Notation

Say:

The result is…

Addition

$a+b$

$a\phantom{\rule{0.2em}{0ex}}\text{plus}\phantom{\rule{0.2em}{0ex}}b$

the sum of $a$ and $b$

Multiplication

$a\cdot b,\left(a\right)\left(b\right),\left(a\right)b,a\left(b\right)$

$a\phantom{\rule{0.2em}{0ex}}\text{times}\phantom{\rule{0.2em}{0ex}}b$

The product of $a$ and $b$

Subtraction

$a-b$

$a\phantom{\rule{0.2em}{0ex}}\text{minus}\phantom{\rule{0.2em}{0ex}}b$

the difference of $a$ and $b$

Division

$a\div b,a/b,\phantom{\rule{0.2em}{0ex}}\frac{a}{b},b\overline{)a}$

$a$ divided by $b$

The quotient of $a$ and $b$

Equality Symbol
- $a=b$ is read as $a$ is equal to $b$
- The symbol $=$ is called the equal sign.
Inequality
- $a$ < $b$ is read $a$ is less than $b$
- $a$ is to the left of $b$ on the number line
- $a$ > $b$ is read $a$ is greater than $b$
- $a$ is to the right of $b$ on the number line

Algebraic Notation	Say
$a=b$	$a$ is equal to $b$
$a\ne b$	$a$ is not equal to $b$
$a$ < $b$	$a$ is less than $b$
$a$ > $b$	$a$ is greater than $b$
$a\le b$	$a$ is less than or equal to $b$
$a\ge b$	$a$ is greater than or equal to $b$

Exponential Notation
- For any expression ${a}^{n}$ is a factor multiplied by itself $n$ times, if $n$ is a positive integer.
- ${a}^{n}$ means multiply $n$ factors of $a$
- The expression of ${a}^{n}$ is read $a$ to the $n\text{th}$ 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.

$9\cdot 6=54$
$5\cdot 4+3$
$x+7$
$y-5=25$

In the following exercises, write in exponential form.

$3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3$
$x\cdot x\cdot x\cdot x\cdot x$

In the following exercises, write in expanded form.

${5}^{3}$
${2}^{8}$

In the following exercises, simplify.

1. $\phantom{\rule{0.2em}{0ex}}3+8\cdot 5\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.4em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}\text{(3+8)}\cdot \text{5}$
${2}^{3}-12 \div \left(9-5\right)$
$3\cdot 8+5\cdot 2$
$2+8\left(6+1\right)$
$4\cdot 12/8$
$6+10/2+2$
$\left(6+10\right)\div \left(2+2\right)$
$20\div 4+6\cdot5$
$20\div \left(4+6\right)\cdot 5$
${4}^{2}+{5}^{2}$
${\left(4+5\right)}^{2}$
$3\left(1+9\cdot 6\right)-{4}^{2}$
$2\left[1+3\left(10-2\right)\right]$

In the following exercises, evaluate the expression for the given value.

$7x+8\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=2$
$5x-4\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=6$
${x}^{2}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=12$
${x}^{2}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=12$
${x}^{2}+3x-7\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=4$
$2x+4y-5\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=7,y=8$
${\left(x-y\right)}^{2}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=10,y=7$
${a}^{2}+{b}^{2}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}a=3,b=8$
$2l+2w\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}l=15,w=12$

Answers:

equation
expression
expression
equation
3⁷
x⁵
125
256
1. 43
2. 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.

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Use Variables and Algebraic Symbols

Identify Expressions and Equations

Simplify Expressions with Exponents

Simplify Expressions Using the Order of Operations

Evaluate Algebraic Expressions

Key Concepts

Glossary

1.1 Exercise Set

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