Order of Operations

When several operations are to be applied within a calculation, we must follow a specific order to ensure a single correct result.

YouTube Playlist on Real Number Operations

Order of Operations:

1. Perform all calculations within the innermost Parentheses first.
2. Evaluate Exponent expressions.
3. Apply the Multiplication and Division operations from left to right.
4. Lastly, work all Addition and Subtraction operations from left to right.

Caution:  Please do not dismiss the fact that multiplication and division should be worked from left to right.  Many standardized exams will test you on this fact.  The following example illustrates the problem.

Order of Operations - The Basics

Simplify.

Order of operation problems get a bit more tedious when fractions are involved. Remember that when adding or subtracting fractions you need to first find the equivalent fractions with a common denominator. Multiplication does not require a common denominator.

Simplify.

The Order of Operations

We will see that some of the problems have different looking parentheses { [ ( ) ] }, treat them the same and just remember to perform the innermost parentheses first.  Some problems may involve an absolute value, in which case you will need to apply the innermost absolute value first as you would if it were a parentheses.

To avoid these unnecessary mistakes, work one operation at a time and for each step rewrite everything.  This may seem like a lot of work but it really helps avoid errors.

Simplify.

Video Examples on YouTube:

Simplifying Algebraic Expressions

The properties of real numbers are very important in our study of Algebra. These properties can be applied in Algebra because a variable is simply a letter that represents a real number. The distributive property is one that we apply often when simplifying algebraic expressions. Given real numbers a, b, c:

a(b + c) = ab + ac

When multiplying an expression within parentheses you must multiply everything inside by the number or variable that you are distributing.

Introduction to the Distributive Property

Simplify.

When simplifying, we will often have to combine like terms after we distribute.  This step is consistent with the order of operations, multiplication before addition.

Combining Like Terms

Simplify.

Simplifying Algebraic Expressions

To combine like terms, the variable parts have to be exactly the same.  But before combining like terms, generally, we will first distribute if necessary.  When distributing negative numbers notice that the operations change.

Simplify.

Profit Word Problem: Profit is equal to revenues less cost of production.  If the revenue R can be represented by

and the cost C can be represented by

where x represents the number of units produced, find an equation that represents the profit.

Subtracting Variable Expressions: What is the difference between 3x − 4 and −2x + 5?

Video Examples on YouTube:

      

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