## Simplify Expressions With the Order of Operations

### Learning Outcomes

• Recognize and combine like terms in an expression
• Use the order of operations to simplify expressions

## Introduction

Some important terminology before we begin:

• operations/operators: In mathematics we call things like multiplication, division, addition, and subtraction operations.  They are the verbs of the math world, doing work on numbers and variables. The symbols used to denote operations are called operators, such as $+{, }-{, }\times{, }\div$. As you learn more math, you will learn more operators.
• term: Examples of terms would be $2x$, $-\dfrac{3}{2}$, or $a^3$. Even lone integers can be a term, like $0$.
• expression: A mathematical expression is one that connects terms with mathematical operators. For example  $\dfrac{1}{2}\normalsize +\left(2^2\right)- 9\div\dfrac{6}{7}$ is an expression.

## Combining Like Terms

One way we can simplify expressions is to combine like terms. Like terms are terms where the variables match exactly (exponents included). Examples of like terms would be $5xy$ and $-3xy$, or $8a^2b$ and $a^2b$, or $-3$ and $8$.  If we have like terms we are allowed to add (or subtract) the numbers in front of the variables, then keep the variables the same. Kind of like saying four apples plus three apples equals seven apples. But two apples plus six oranges can’t be combined and simplified because they are not “like terms”. As we combine like terms we need to interpret subtraction signs as part of the following term. This means if we see a subtraction sign, we treat the following term like a negative term. The sign always stays with the term.

This is shown in the following examples:

### Example

Combine like terms:  $5x-2y-8x+7y$

In the following video you will be shown how to combine like terms using the idea of the distributive property.  Note that this is a different method than is shown in the written examples on this page, but it obtains the same result.

### Example

Combine like terms:  $x^2-3x+9-5x^2+3x-1$

In the video that follows, you will be shown another example of combining like terms.  Pay attention to why you are not able to combine all three terms in the example.

## Order of Operations

You may or may not recall the order of operations for applying several mathematical operations to one expression. Just as it is a social convention for us to drive on the right-hand side of the road, the order of operations is a set of conventions used to provide a formulaic outline to follow when you are required to use several mathematical operations for one expression.

### The Order of Operations

• Perform all operations within grouping symbols first. Grouping symbols include parentheses ( ), brackets [ ], braces { }, and fraction bars.
• Evaluate exponents or square roots.
• Multiply or divide, from left to right.
• Add or subtract, from left to right.

This order of operations is true for all real numbers.

### Example

Simplify $7–5+3\cdot8$

In the following example, you will be shown how to simplify an expression that contains both multiplication and subtraction using the order of operations.

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.

### Example

Simplify $3\cdot\dfrac{1}{3}\normalsize -8\div\dfrac{1}{4}$

In the following video you are shown how to use the order of operations to simplify an expression that contains multiplication, division, and subtraction with terms that contain fractions.

## Exponents and Square Roots

In this section, we expand our skills with applying the order of operation rules to expressions with exponents and square roots. If the expression has exponents or square roots, they are to be performed after parentheses and other grouping symbols have been simplified and before any multiplication, division, subtraction, and addition that are outside the parentheses or other grouping symbols.

Recall that an expression such as $7^{2}$ is exponential notation for $7\cdot7$. Exponential notation has two parts: the base and the exponent or the power. In $7^{2}$, $7$ 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.

### Example

Simplify $3^{2}\cdot2^{3}$

In the video that follows, an expression with exponents on its terms is simplified using the order of operations.

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.

### Example

Simplify $\left(3+4\right)^{2}+\left(8\right)\left(4\right)$

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.

In the next example we will simplify an expression that has a square root.

### Example

Simplify $\dfrac{\sqrt{7+2}+2^2}{(8)(4)-11}$

These problems are very similar to the examples given above. How are they different and what tools do you need to simplify them?

a) Simplify $\left(1.5+3.5\right)–2\left(0.5\cdot6\right)^{2}$. This problem has parentheses, exponents, multiplication, subtraction, and addition in it, as well as decimals instead of integers.

Use the box below to write down a few thoughts about how you would simplify this expression with decimals and grouping symbols.

b) Simplify ${{\left(\dfrac{1}{2}\normalsize\right)}^{2}}+{{\left(\dfrac{1}{4}\normalsize\right)}^{3}}\cdot \,32$

Use the box below to write down a few thoughts about how you would simplify this expression with fractions and grouping symbols.

Some people use a saying to help them remember the order of operations. This saying is called PEMDAS or Please Excuse My Dear Aunt Sally. The first letter of each word begins with the same letter of an arithmetic operation.

Please $\displaystyle \Rightarrow$ Parentheses (and other grouping symbols)
Excuse $\displaystyle \Rightarrow$ Exponents
My Dear $\displaystyle \Rightarrow$ Multiplication and Division (from left to right)
Aunt Sally $\displaystyle \Rightarrow$ Addition and Subtraction (from left to right)

Note: Even though multiplication comes before division in the saying, division could be performed first. Which is performed first, between multiplication and division, is determined by which comes first when reading from left to right. The same is true of addition and subtraction. Don’t let the saying confuse your understanding of this!

## Summary

The order of operations gives us a consistent sequence to use in computation. Without the order of operations, you could come up with different answers to the same computation problem. (Some of the early calculators, and some inexpensive ones, do NOT use the order of operations. In order to use these calculators, the user has to input the numbers in the correct order.)