0.4 Simplifying Expressions and Order of Operations
Order of Operations
Order of operations is a set of conventions used to provide order when you are required to use several mathematical operations for one expression. You may recall one way to remember order of operations is the phrase “Please Excuse My Dear Aunt Sally” for Parentheses, Exponents, Multiplication/Division, and Addition/Subtraction.
Order of operations
Perform all operations within grouping symbols first, including {}, [], and ().
Evaluate exponents or square roots.
Multiply or divide from left to right.
Add or subtract from left to right.
Example
Simplify [latex]7–5+3\cdot8[/latex].
Show Solution
According to the order of operations, multiplication comes before addition and subtraction.
Now, add and subtract from left to right. [latex]7–5[/latex] comes first.
[latex]2+24[/latex].
Finally, add.
[latex]2+24=26[/latex]
Answer
[latex]7–5+3\cdot8=26[/latex]
When you are applying the order of operations to expressions that contain fractions, decimals, and negative numbers, you will need to recall how to do these computations as well.
According to the order of operations, multiplication and division come before addition and subtraction. Sometimes it helps to add parentheses to help you know what comes first, so let’s put parentheses around the multiplication and division since it will come before the subtraction.
When you are evaluating expressions, you will sometimes see exponents used to represent repeated multiplication. Recall that an expression such as [latex]7^{2}[/latex] is exponential notation for [latex]7\cdot7[/latex]. (Exponential notation has two parts: the base and the exponent or the power. In [latex]7^{2}[/latex], 7 is the base and 2 is the exponent; the exponent determines how many times the base is multiplied by itself.)
Exponents are a way to represent repeated multiplication; the order of operations places it before any other multiplication, division, subtraction, and addition is performed. The next section of the math review goes into detail about exponent rules. Examples of orders of operations involving exponents will appear in the next page titled “Exponents”
Grouping Symbols
Grouping symbols such as parentheses ( ), brackets [ ], braces[latex] \displaystyle \left\{ {} \right\}[/latex], and fraction bars can be used to further control the order of the four arithmetic operations. The rules of the order of operations require computation within grouping symbols to be completed first, even if you are adding or subtracting within the grouping symbols and you have multiplication outside the grouping symbols. After computing within the grouping symbols, divide or multiply from left to right and then subtract or add from left to right. When there are grouping symbols within grouping symbols, calculate from the inside to the outside. That is, begin simplifying within the innermost grouping symbols first.
Remember that parentheses can also be used to show multiplication. In the example that follows, both uses of parentheses—as a way to represent a group, as well as a way to express multiplication—are shown.
This problem has parentheses, exponents, multiplication, and addition in it. The first set of parentheses is a grouping symbol. The second set indicates multiplication.
Grouping symbols are handled first. Add numbers in parentheses.
In this section, we will use the skills from the last section to simplify mathematical expressions that contain many grouping symbols and many operations. We are using the term compound to describe expressions that have many operations and many grouping symbols. More care is needed with these expressions when you apply the order of operations. Additionally, you will see how to handle absolute value terms when you simplify expressions.
This problem has brackets, parentheses, fractions, exponents, multiplication, subtraction, and addition in it.
Grouping symbols are handled first. The parentheses around the [latex]-6[/latex] aren’t a grouping symbol; they are simply making it clear that the negative sign belongs to the 6. Start with the innermost set of parentheses that are a grouping symbol. In this example, the innermost set of parentheses would be in the numerator of the fraction, [latex](2\cdot(−6))[/latex]. Begin working out from there. (The fraction line acts as a type of grouping symbol, too; you simplify the numerator and denominator independently, and then divide the numerator by the denominator at the end.)
The top of the fraction is all set, but the bottom (denominator) has remained untouched. Apply the order of operations to that as well. Begin by evaluating [latex]3^{2}=9[/latex].
Parentheses are used to group or combine expressions and terms in mathematics. You may see them used when you are working with formulas, and when you are translating a real situation into a mathematical problem so you can find a quantitative solution.
The following definition describes how to use the distributive property in general terms.
The Distributive Property of Multiplication
For all real numbers a, b, and c, [latex]a(b+c)=ab+ac[/latex].
What this means is that when a number multiplies an expression inside parentheses, you can distribute the multiplication to each term of the expression individually.
To simplify [latex]3\left(3+y\right)-y+9[/latex], it may help to see the expression translated into words:
multiply three by (the sum of three and y), then subtract y, then add 9
To multiply three by the sum of three and y, you use the distributive property –
Absolute value expressions are one final method of grouping that you may see. Recall that the absolute value of a quantity is always positive or 0.
When you see an absolute value expression included within a larger expression, treat the absolute value like a grouping symbol and evaluate the expression within the absolute value sign first. Then take the absolute value of that expression. The example below shows how this is done.
