Use the Language of Algebra

Read this text. Pay special attention to the Order of Operations portion that summarizes the rules for performing multi-step calculations.

Many students use a silly phrase to remember the order of operations: Please Excuse My Dear Aunt Sally, where the first letter of each word corresponds to a mathematical operation: parentheses, exponents, multiplication and division, addition and subtraction.

Do Examples 2.8 – 2.12, and check your answers. If you need more practice, you can try Try It 2.15 – 2.24.

Solutions

Example 2.8

Simplify the expressions:
ⓐ 4+3⋅7
ⓑ (4+3)⋅7

Solution

ⓐ 4+3⋅7
Are there any parentheses? No.
Are there any exponents? No.
Is there any multiplication or division? Yes.
Multiply first. 4+3⋅7
Add. 4+21
25
ⓑ
(4+3)⋅7
Are there any parentheses? Yes.
(4+3)⋅7
Simplify inside the parentheses.  (7)7
Are there any exponents? No
Is there any multiplication or division? Yes.
Multiply 49


Example 2.9

Simplify:
ⓐ 18÷9⋅2
ⓑ 18⋅9÷2

Solution

ⓐ 18÷9⋅2
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.  2 \cdot 2
Multiply. 4
ⓑ
18⋅9÷2
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. 162 ÷ 2
Divide. 81


Example 2.10

Simplify: 18÷6+4(5−2).

Solution

18÷6+4(5−2)
Parentheses? Yes, subtract first.
 18÷6+4(3)
Exponents? No.
Multiplication or division? Yes.
Divide first because we multiply and divide left to right.
 3+4(3)
Any other multiplication or division? Yes.
Multiply.
 3+12
Any other multiplication or division? No.
Any addition or subtraction? Yes.
 15


Example 2.11

Simplify: 5+2^3+3[6−3(4−2)].

Solution

5+2^3+3[6−3(4−2)]
Are there any parentheses (or other grouping symbol)? Yes.

Focus on the parentheses that are inside the brackets.
 5+2^3+3[6−3(4−2)]
Subtract.
 5+2^3+3[6−3(2)]
Continue inside the brackets and multiply.
 5+2^3+3[6−6]
Continue inside the brackets and subtract.
 5+2^3+3[0]
The expression inside the brackets requires no further simplification.
Are there any exponents? Yes.
Simplify exponents.
 5+2^3+3[0]
Is there any multiplication or division? Yes.
Multiply.
 5+8+3[0]
Is there any addition or subtraction? Yes.
Add.
 5+8+0
Add.
 13+0
13


Example 2.12

Simplify: 2^3+3^4÷3−5^2.

Solution

2^3+3^4÷3−5^2
If an expression has several exponents, they may be simplified in the same step.
Simplify exponents.
 2^3+3^4÷3−5^2
Divide.
 8+81÷3-25
Add.
 8+27-25
Subtract. 35-25
10

Try It 2.15
    ⓐ 2
    ⓑ 14

Try It 2.16
    ⓐ 35
    ⓑ 99

Try It 2.17
18

Try It 2.18
9

Try It 2.19
16

Try It 2.20
23

Try It 2.21
86

Try It 2.22
1

Try It 2.23
81

Try It 2.24
75