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The Square of a Binomial
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The Pythagorean Theorem
Solving Radical Equations in One Variable
Multiplying Binomials Using the FOIL Method
Imaginary Numbers
Solving Quadratic Equations Using the Quadratic Formula
Solving Quadratic Equations
Algebra
Order of Operations
Dividing Complex Numbers
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The Appearance of a Polynomial Equation
Standard Form of a Line
Positive Integral Divisors
Dividing Fractions
Solving Linear Systems of Equations by Elimination
Factoring
Multiplying and Dividing Square Roots
Functions and Graphs
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Solving Rational Equations
Numbers
Use of Parentheses or Brackets (The Distributive Law)
Multiplying and Dividing by Monomials
Solving Quadratic Equations by Graphing
Multiplying Decimals
Use of Parentheses or Brackets (The Distributive Law)
Simplifying Complex Fractions 1
Adding Fractions
Simplifying Complex Fractions
Solutions to Linear Equations in Two Variables
Quadratic Expressions Completing Squares
Dividing Radical Expressions
Rise and Run
Graphing Exponential Functions
Multiplying by a Monomial
The Cartesian Coordinate System
Writing the Terms of a Polynomial in Descending Order
Fractions
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Solving Inequalities
Solving Rational Inequalities with a Sign Graph
Solving Linear Equations
Solving an Equation with Two Radical Terms
Simplifying Rational Expressions
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Intercepts of a Line
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Order of Operations
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Collecting Like Terms
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Percent of Change
Powers of ten (Scientific Notation)
Comparing Integers on a Number Line
Solving Systems of Equations Using Substitution
Factoring Out the Greatest Common Factor
Families of Functions
Monomial Factors
Multiplying and Dividing Complex Numbers
Properties of Exponents
Multiplying Square Roots
Radicals
Adding or Subtracting Rational Expressions with Different Denominators
Expressions with Variables as Exponents
The Quadratic Formula
Writing a Quadratic with Given Solutions
Simplifying Square Roots
Adding and Subtracting Square Roots
Adding and Subtracting Rational Expressions
Combining Like Radical Terms
Solving Systems of Equations Using Substitution
Dividing Polynomials
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Product of a Sum and a Difference
Solving First Degree Inequalities
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Roots and Powers
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Order of Operations

When an expression contains more than one operation, we must decide which operation to carry out first. To do so, we use the following procedure.

 

Procedure

To Use the Order of Operations to Simplify an Expression

Step 1 Perform operations inside grouping symbols (such as parentheses), starting with the innermost set of grouping symbols.

Step 2 Simplify exponents, square roots, and absolute values.

Step 3 Multiply or divide, working in order from left to right.

Step 4 Add or subtract, working in order from left to right.

 

The grouping symbols referred to in Step 1 include the following:

Symol Name Example
( ) parentheses 6 ÷ (2 + 1) = 6 ÷ 3 = 2
[ ] brackets 12 - [8 - (4 - 1)] = 12 - [8 - 3] = 12 - 5 = 7
fraction bar
|  | absolute value 3 · |1 - 5 | = 3 · |-4| = 3 · 4 = 12
radical symbol

 

Example 1

Find 5 - 62 ÷ 2 - (9 - 5) · 3

 

Solution 5 - 62 ÷ 2 - (9 - 5) · 3
Step 1 Perform operations inside grouping symbols. = 5 - 62 ÷ 2 - 4 · 3
Step 2 Simplify exponents, square roots, and absolute values. = 5 - 36 ÷ 2 - 4 · 3
Step 3 Multiply or divide, working in order from left to right. = 5 - 18 - 12
Step 4 Add or subtract, working in order from left to right. = -25
So, the result is -25.

 

Example 2

Find:

Solution
Step 1 Perform operations inside grouping symbols.
Step 2 Simplify exponents, square roots,and absolute values. = 3 + 16 ÷ 2
Step 3 Multiply or divide, working in order from left to right. = 3 + 8
Step 4 Add or subtract, working in order from left to right. = 11
Thus, the result is 11.

 

Example 3

Find 20 ÷ 2 [1 - (8 - 12)]

Solution 20 ÷ 2 [1 - (8 - 12)]
Step 1 Perform operations inside grouping symbols, starting with the innermost grouping symbol.  
First, simplify (8 - 12).

Now simplify [1 - (-4)].

= 20 ÷ 2[1- (-4)]

= 20 ÷ 2[5]

= 20 ÷ 2 · 5

Step 2 Simplify exponents, square roots, and absolute values.

There are none to simplify.

 
Step 3 Multiply or divide, working in order from left to right.  
First, simplify 20 ÷ 2.

Now simplify 10 · 5.

= 10 · 5

= 50

 Therefore, the result is 50.
 
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