Learning Outcomes
- Use the order of operations to evaluate expressions with real numbers
Introduction
Let’s review some important terminology before we begin:
A mathematical expression combines numbers and variables with mathematical operations such as addition, subtraction, multiplication, and addition. For example, [latex]2+8 \cdot 5[/latex] is an expression.
We obtain different results depending on the order in which we perform the operations in an expression. We need a set of conventions so that everyone arrives at the same value when evaluating an expression.
Order of Operations
- Perform all operations within grouping symbols first. Grouping symbols include ( ), [ ], and { }. If there are nested groupings, evaluate within the innermost grouping symbols first.
- Evaluate exponents and radicals (such as square roots)
- Multiply and divide, left to right
- Add and subtract, left to right
A helpful acronym for remembering the order of operations is PEMDAS (parenthesis, exponents, multiplications and divisions, additions, and subtractions).
Example
Simplify [latex]2 + 8 \cdot 5[/latex].
Example
Simplify [latex]10-6 \div 3 \cdot 2 +1[/latex].
In the following example, you will be shown how to simplify an expression that contains both multiplication and subtraction using the order of operations.
https://youtu.be/yqp06obmcVc
Exponents
When you are evaluating expressions, you will sometimes see exponents used to represent repeated multiplication. Recall that an expression such as [latex]2^{3}[/latex] is exponential notation for [latex]2 \cdot 2 \cdot 2[/latex]. Exponential notation has two parts: the base and the exponent or the power. In [latex]2^{3}, 2[/latex] is the base, and [latex]3[/latex] 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.
Example
Simplify [latex]5 \cdot 2^{3}[/latex]
In the video that follows, an expression with exponents on its terms is simplified using the order of operations.
Grouping Symbols
Grouping symbols such as parentheses ( ), brackets [ ], braces{ }, and fraction bars can be used to further control the order of the four arithmetic operations. The rules of the order of operations require computations within grouping symbols to be completed first. As you evaluate within a grouping symbol, follow the order of operations. 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.
Grouping symbols 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.
Example
Simplify [latex][12-(1+3^{2})](5)[/latex]
Example
Simplify [latex] \frac{2+7}{4-1}[/latex]
In the following video, you are shown how to use the order of operations to simplify an expression with grouping symbols, exponents, multiplication, and addition.
Try It
Often we will need to evaluate a formula for given values of the variables. When we do so, we replace each variable with its given value and evaluate according to the order of operations. For example, the volume of a square pyramid can be found by evaluating [latex]V=\frac{1}{3}s^{2}h[/latex], where [latex]V[/latex] represents volume, [latex]s[/latex] is the length of the sides of the square base, and [latex]h[/latex] is the height. If [latex]s=12[/latex] centimeters and [latex]h=10[/latex] centimeters, the volume of the square pyramid is
[latex]V=\frac{1}{3} \cdot 12^{2} \cdot 10 = \frac{1}{3} \cdot 144 \cdot 10 = \frac{144}{3} \cdot 10 = 48 \cdot 10 = 480[/latex] cubic centimeters.
