CHAPTER 1 Whole Numbers, Integers, and Introduction to Algebra

1.2 Use the Language of Algebra

Learning Objectives

By the end of this section, 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

Use Variables and Algebraic Symbols

Greg and Alex have the same birthday, but they were born in different years. This year Greg is 20 years old and Alex is 23, so Alex is 3 years older than Greg. When Greg was 12, Alex was 15. When Greg is 35, Alex will be 38. No matter what Greg’s age is, Alex’s age will always be 3 years more, right?

In the language of algebra, we say that Greg’s age and Alex’s age are variable and the three is a constant. The ages change, or vary, so age is a variable. The 3 years between them always stays the same, so the age difference is the constant.

In algebra, letters of the alphabet are used to represent variables. Suppose we call Greg’s age g. Then we could use g+3 to represent Alex’s age. See the table below.

Greg’s age Alex’s age
12 15
20 23
35 38
g g+3

Letters 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. In 1.1 Whole Numbers, we introduced the symbols for the 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.

We perform these operations on two numbers. When translating from symbolic form to words, or from words to symbolic form, pay attention to the words of or and to help you find the numbers.

  • The sum of 5 and 3 means add 5 plus 3, which we write as 5+3.
  • The difference of 9 and 2 means subtract 9 minus 2, which we write as 9-2.
  • The product of 4 and 8 means multiply 4 times 8, which we can write as 4\cdot 8.
  • The quotient of 20 and 5 means divide 20 by 5, which we can write as 20 \div 5.

EXAMPLE 1

Translate from algebra to words:

  1. \phantom{\rule{0.2em}{0ex}}12+14\phantom{\rule{0.2em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}\left(30\right)\left(5\right)\phantom{\rule{0.2em}{0ex}}
  3. \phantom{\rule{0.2em}{0ex}}64 \div 8\phantom{\rule{0.2em}{0ex}}
  4. \phantom{\rule{0.2em}{0ex}}x-y
Solution
a.
12+14
12 plus 14
the sum of twelve and fourteen
b.
\left(30\right)\left(5\right)
30 times 5
the product of thirty and five
c.
64\div 8
64 divided by 8
the quotient of sixty-four and eight
d.
x-y
x minus y
the difference of x and y

TRY IT 1.1

Translate from algebra to words.

  1. \phantom{\rule{0.2em}{0ex}}18+11\phantom{\rule{0.4em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}\left(27\right)\left(9\right)\phantom{\rule{0.4em}{0ex}}
  3. \phantom{\rule{0.2em}{0ex}}84\div 7
  4. p-q
Show Answer
  1. 18 plus 11; the sum of eighteen and eleven
  2. 27 times 9; the product of twenty-seven and nine
  3. 84 divided by 7; the quotient of eighty-four and seven
  4. p minus q; the difference of p and q

TRY IT 1.2

Translate from algebra to words.

  1. \phantom{\rule{0.2em}{0ex}}47-19\phantom{\rule{0.4em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}72\div 9\phantom{\rule{0.4em}{0ex}}
  3. \phantom{\rule{0.2em}{0ex}}m+n\phantom{\rule{0.4em}{0ex}}
  4. \phantom{\rule{0.2em}{0ex}}\left(13\right)\left(7\right)
Show Answer
  1. 47 minus 19; the difference of forty-seven and nineteen
  2. 72 divided by 9; the quotient of seventy-two and nine
  3. m plus n; the sum of m and n
  4. 13 times 7; the product of thirteen and seven

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.

EXAMPLE 2

Translate from algebra to words:

  1. \phantom{\rule{0.2em}{0ex}}20\le 35
  2. \phantom{\rule{0.2em}{0ex}}11\ne 15-3
  3. \phantom{\rule{0.2em}{0ex}}9 > \phantom{\rule{0.2em}{0ex}}10\div 2
  4. \phantom{\rule{0.2em}{0ex}}x+2 < \phantom{\rule{0.2em}{0ex}}10
Solution
a.
20\le 35
20 is less than or equal to 35
b.
11\ne 15-3
11 is not equal to 15 minus 3
c.
9 > 10\div 2
9 is greater than 10 divided by 2
d.
x+2 < 10
x plus 2 is less than 10

TRY IT 2.1

Translate from algebra to words.

  1. \phantom{\rule{0.2em}{0ex}}\text{14}\le 27\phantom{\rule{0.2em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}19-2\ne 8\phantom{\rule{0.2em}{0ex}}
  3. \phantom{\rule{0.2em}{0ex}}12 > 4\phantom{\rule{0.2em}{0ex}}\div 2\phantom{\rule{0.2em}{0ex}}
  4. \phantom{\rule{0.2em}{0ex}}x-7 < \phantom{\rule{0.2em}{0ex}}1
Show Answer
  1. fourteen is less than or equal to twenty-seven
  2. nineteen minus two is not equal to eight
  3. twelve is greater than four divided by two
  4. x minus seven is less than one

TRY IT 2.2

Translate from algebra to words.

  1. \phantom{\rule{0.2em}{0ex}}19\ge 15\phantom{\rule{0.2em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}7=12-5\phantom{\rule{0.2em}{0ex}}
  3. \phantom{\rule{0.2em}{0ex}}15\div 3 < 8\phantom{\rule{0.2em}{0ex}}
  4. \phantom{\rule{0.2em}{0ex}}y-3 > 6\phantom{\rule{0.2em}{0ex}}
Show Answer
  1. nineteen is greater than or equal to fifteen
  2. seven is equal to twelve minus five
  3. fifteen divided by three is less than eight
  4. y minus three is greater than six

EXAMPLE 3

The information in (Figure 1) compares the fuel economy in miles-per-gallon (mpg) of several cars. Write the appropriate symbol \text{symbol},\text{=}, < ,\text{or} >. in each expression to compare the fuel economy of the cars.

(credit: modification of work by Bernard Goldbach, Wikimedia Commons)
This table has two rows and six columns. The first column is a header column and it labels each row The first row is labeled “Car” and the second “Fuel economy (mpg)”. To the right of the ‘Car’ row are the labels: “Prius”, “Mini Cooper”, “Toyota Corolla”, “Versa”, “Honda Fit”. Each of these columns contains an image of the labeled car model. To the right of the “Fuel economy (mpg)” row are the algebraic equations: the letter p, the equals symbol, the number forty-eight; the letter m, the equals symbol, the number twenty-seven; the letter c, the equals symbol, the number twenty-eight; the letter v, the equals symbol, the number twenty-six; and the letter f, the equals symbol, the number twenty-seven.
Figure 1
  1. MPG of Prius_____ MPG of Mini Cooper
  2. MPG of Versa_____ MPG of Fit
  3. MPG of Mini Cooper_____ MPG of Fit
  4. MPG of Corolla_____ MPG of Versa
  5. MPG of Corolla_____ MPG of Prius
Solution
a.
MPG of Prius____MPG of Mini Cooper
Find the values in the chart. 48____27
Compare. 48 > 27
MPG of Prius > MPG of Mini Cooper
b.
MPG of Versa____MPG of Fit
Find the values in the chart. 26____27
Compare. 26 < 27
MPG of Versa < MPG of Fit
c.
MPG of Mini Cooper____MPG of Fit
Find the values in the chart. 27____27
Compare. 27 = 27
MPG of Mini Cooper = MPG of Fit
d.
MPG of Corolla____MPG of Versa
Find the values in the chart. 28____26
Compare. 28 > 26
MPG of Corolla > MPG of Versa
e.
MPG of Corolla____MPG of Prius
Find the values in the chart. 28____48
Compare. 28 < 48
MPG of Corolla < MPG of Prius

TRY IT 3.1

Use Figure 1 to fill in the appropriate \text{symbol},\text{=}, < ,\text{or} >.

  1. MPG of Prius_____MPG of Versa
  2. MPG of Mini Cooper_____ MPG of Corolla
Show Answer
  1. >
  2. <

TRY IT 3.2

Use Figure 1 to fill in the appropriate \text{symbol},\text{=}, < ,\text{or} >.

  1. MPG of Fit_____ MPG of Prius
  2. MPG of Corolla _____ MPG of Fit
Show Answer
  1. <
  2. <

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
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 4

Determine if each is an expression or an equation:

  1. \phantom{\rule{0.2em}{0ex}}16-6=10
  2. \phantom{\rule{0.2em}{0ex}}4\cdot 2+1
  3. \phantom{\rule{0.2em}{0ex}}x\div 25
  4. \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 4.1

Determine if each is an expression or an equation:

  1. \phantom{\rule{0.2em}{0ex}}23+6=29\phantom{\rule{0.4em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}7\cdot 3-7
Show Answer
  1. equation
  2. expression

TRY IT 4.2

Determine if each is an expression or an equation:

  1. \phantom{\rule{0.2em}{0ex}}y\div 14\phantom{\rule{0.4em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}x-6=21
Show Answer
  1. expression
  2. equation

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 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

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 5

Write each expression in exponential form:

  1. \phantom{\rule{0.2em}{0ex}}16\cdot16\cdot16\cdot16\cdot16\cdot16\cdot16
  2. \phantom{\rule{0.2em}{0ex}}9\cdot9\cdot9\cdot9\cdot9
  3. \phantom{\rule{0.2em}{0ex}}x\cdot x\cdot x\cdot x
  4. \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 5.1

Write each expression in exponential form:

41\cdot41\cdot41\cdot41\cdot41

Show Answer

415

TRY IT 5.2

Write each expression in exponential form:

7\cdot7\cdot7\cdot7\cdot7\cdot7\cdot7\cdot7\cdot7

Show Answer

79

EXAMPLE 6

Write each exponential expression in expanded form:

  1. \phantom{\rule{0.2em}{0ex}}{8}^{6}\phantom{\rule{0.2em}{0ex}}
  2. \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 6.1

Write each exponential expression in expanded form:

  1. \phantom{\rule{0.2em}{0ex}}{4}^{8}
  2. \phantom{\rule{0.2em}{0ex}}{a}^{7}
Show Answer
  1. 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4
  2. a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a

TRY IT 6.2

Write each exponential expression in expanded form:

  1. {\phantom{\rule{0.2em}{0ex}}8}^{8}\phantom{\rule{0.4em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}{b}^{6}
Show Answer
  1. 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8
  2. b \cdot b \cdot b \cdot b \cdot b \cdot b

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

EXAMPLE 7

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 7.1

Simplify:

  1. \phantom{\rule{0.2em}{0ex}}{5}^{3}\phantom{\rule{0.4em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}{1}^{7}
Show Answer
  1. 125
  2. 1

TRY IT 7.2

Simplify:

  1. \phantom{\rule{0.2em}{0ex}}{7}^{2}\phantom{\rule{0.4em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}{0}^{5}
Show Answer
  1. 49
  2. 0

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 8

Simplify the expressions:

  1. \phantom{\rule{0.2em}{0ex}}4+3\cdot 7\phantom{\rule{0.2em}{0ex}}
  2. \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 8.1

Simplify the expressions:

  1. \phantom{\rule{0.2em}{0ex}}12-5\cdot 2\phantom{\rule{0.4em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}\left(12-5\right)\cdot 2
Show Answer
  1. 2
  2. 14

TRY IT 8.2

Simplify the expressions:

  1. \phantom{\rule{0.2em}{0ex}}8+3\cdot 9\phantom{\rule{0.4em}{0ex}}
  2. \left(8+3\right)\cdot 9
Show Answer
  1. 35
  2. 99

EXAMPLE 9

Simplify:

  1. \phantom{\rule{0.2em}{0ex}}\text{18}\div \text{9}\cdot \text{2}\phantom{\rule{0.2em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}\text{18}\cdot \text{9}\div \text{2}
Solution
a.
.
Are there any parentheses? No.
Are there any exponents? No.
Is there any multiplication or division? Yes.
Multiply and divide from left to right. Divide. .
Multiply. .
b.
.
Are there any parentheses? No.
Are there any exponents? No.
Is there any multiplication or division? Yes.
Multiply and divide from left to right.
Multiply. .
Divide. .

TRY IT 9.1

Simplify:

42\div 7\cdot 3

Show Answer

18

TRY IT 9.2

Simplify:

12\cdot 3\div 4

Show Answer

9

EXAMPLE 10

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 10.1

Simplify:

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

Show Answer

16

TRY IT 10.2

Simplify:

70\div 10+4\left(6-2\right)

Show Answer

23

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

EXAMPLE 11

\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 11.1

Simplify:

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

Show Answer

86

TRY IT 11.2

Simplify:

{7}^{2}-2\left[4\left(5+1\right)\right]

Show Answer

1

EXAMPLE 12

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 12.1

Simplify:

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

Show Answer

81

TRY IT 12.2

Simplify:

{6}^{2}-{5}^{3}\div 5+{8}^{2}

Show Answer

75

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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.

Practice Makes Perfect

Use Variables and Algebraic Symbols

In the following exercises, translate from algebraic notation to words.

1. 16-9 2. 25-7
3. 5 \cdot 6 4. 3 \cdot 9
5. 28 \div 4 6. 45 \div 5
7. x+8 8. x+11
9. \left(2\right)\left(7\right) 10. \left(4\right)\left(8\right)
11. 14 < 21 12. 17 < 35
13. 36\ge 19 14. 42\ge 27
15. 3n=24 16. 6n=36
17. y-1 > 6 18. y-4 > 8
19. 2\le 18 \div 6 20. 3\le 20 \div 4
21. a\ne 7\cdot 4 22. a\ne 1\cdot 12

Identify Expressions and Equations

In the following exercises, determine if each is an expression or an equation.

23. 9\cdot 6=54 24. 7\cdot 9=63
25. 5\cdot 4+3 26. 6\cdot 3+5
27. x+7 28. x+9
29. y-5=25 30. y-8=32

Simplify Expressions with Exponents

In the following exercises, write in exponential form.

31. 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3 32. 4\cdot 4\cdot 4\cdot 4\cdot 4\cdot 4
33. x\cdot x\cdot x\cdot x\cdot x 34. y\cdot y\cdot y\cdot y\cdot y\cdot y

In the following exercises, write in expanded form.

35. {5}^{3} 36. {8}^{3}
37. {2}^{8} 38. {10}^{5}

Simplify Expressions Using the Order of Operations

In the following exercises, simplify.

39.

a.\phantom{\rule{0.2em}{0ex}}3+8\cdot 5\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.4em}{0ex}}

b.\phantom{\rule{0.2em}{0ex}}\text{(3+8)}\cdot \text{5}

 40.

a.\phantom{\rule{0.2em}{0ex}}2+6\cdot 3\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.4em}{0ex}}

b.\phantom{\rule{0.2em}{0ex}}\text{(2+6)}\cdot \text{3}
41. {2}^{3}-12 \div \left(9-5\right) 42. {3}^{2}-18\div \left(11-5\right)
43. 3\cdot 8+5\cdot 2 44. 4\cdot 7+3\cdot 5
45. 2+8\left(6+1\right) 46. 4+6\left(3+6\right)
47. 4\cdot 12/8 48. 2\cdot 36/6
49. 6+10/2+2 50. 9+12/3+4
51. \left(6+10\right)\div \left(2+2\right) 52. \left(9+12\right)\div \left(3+4\right)
53. 20\div 4+6\cdot5 54. 33\div 3+8\cdot2
55. 20\div \left(4+6\right)\cdot 5 56. 33\div \left(3+8\right)\cdot 2
57. {4}^{2}+{5}^{2} 58. {3}^{2}+{7}^{2}
59. {\left(4+5\right)}^{2} 60. {\left(3+7\right)}^{2}
61. 3\left(1+9\cdot 6\right)-{4}^{2} 62. 5\left(2+8\cdot 4\right)-{7}^{2}
63. 2\left[1+3\left(10-2\right)\right] 64. 5\left[2+4\left(3-2\right)\right]

Everyday Math

65. Basketball In the 2014 NBA playoffs, the San Antonio Spurs beat the Miami Heat. The table below shows the heights of the starters on each team. Use this table to fill in the appropriate symbol ( = ,<,  >).

Spurs Height Heat Height
Tim Duncan 83'' Rashard Lewis 82''
Boris Diaw 80'' LeBron James 80''
Kawhi Leonard 79'' Chris Bosh 83''
Tony Parker 74'' Dwyane Wade 76''
Danny Green 78'' Ray Allen 77''
  1. Height of Tim Duncan____Height of Rashard Lewis
  2. Height of Boris Diaw____Height of LeBron James
  3. Height of Kawhi Leonard____Height of Chris Bosh
  4. Height of Tony Parker____Height of Dwyane Wade
  5. Height of Danny Green____Height of Ray Allen

66. Elevation In Colorado there are more than 50 mountains with an elevation of over 14,000\phantom{\rule{0.2em}{0ex}}\text{feet.} The table shows the ten tallest. Use this table to fill in the appropriate inequality symbol.

Mountain Elevation
Mt. Elbert 14,433'
Mt. Massive 14,421'
Mt. Harvard 14,420'
Blanca Peak 14,345'
La Plata Peak 14,336'
Uncompahgre Peak 14,309'
Crestone Peak 14,294'
Mt. Lincoln 14,286'
Grays Peak 14,270'
Mt. Antero 14,269'
  1. Elevation of La Plata Peak____Elevation of Mt. Antero
  2. Elevation of Blanca Peak____Elevation of Mt. Elbert
  3. Elevation of Gray’s Peak____Elevation of Mt. Lincoln
  4. Elevation of Mt. Massive____Elevation of Crestone Peak
  5. Elevation of Mt. Harvard____Elevation of Uncompahgre Peak

Writing Exercises

67.Explain the difference between an expression and an equation. 68. Why is it important to use the order of operations to simplify an expression?

Answers:

1. 16 minus 9, the difference of sixteen and nine 3. 5 times 6, the product of five and six 5. 28 divided by 4, the quotient of twenty-eight and four
7. x plus 8, the sum of x and eight 9. 2 times 7, the product of two and seven 11. fourteen is less than twenty-one
13. thirty-six is greater than or equal to nineteen 15. 3 times n equals 24, the product of three and n equals twenty-four 17. y minus 1 is greater than 6, the difference of y and one is greater than six
19. 2 is less than or equal to 18 divided by 6; 2 is less than or equal to the quotient of eighteen and six 21. a is not equal to 7 times 4, a is not equal to the product of seven and four 23. equation
25. expression 27. expression 29. equation
31. 37 33. x5 35. 125
37. 256 39.

a. 43

b. 55

 41. 5
43. 34 45. 58 47. 6
49. 13 51. 4 53. 35
55. 10 57. 41 59. 81
61. 149 63. 50 65. a. > b. = c. < d. < e. >
67. Answer may vary.

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.

License

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Introductory Algebra Copyright © 2021 by Izabela Mazur is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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