Simplifying Expressions Using the Order of Operations

Learning Outcomes

Use the order of operations to simplify mathematical expressions
Simplify mathematical expressions involving addition, subtraction, multiplication, division, and exponents

Simplify Expressions Using the Order of Operations

We’ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values.

For example, consider the expression:

[latex]4+3\cdot 7[/latex]

[latex]\begin{array}{cccc}\hfill \text{Some students say it simplifies to 49.}\hfill & & & \hfill \text{Some students say it simplifies to 25.}\hfill \\ \begin{array}{ccc}& & \hfill 4+3\cdot 7\hfill \\ \text{Since }4+3\text{ gives 7.}\hfill & & \hfill 7\cdot 7\hfill \\ \text{And }7\cdot 7\text{ is 49.}\hfill & & \hfill 49\hfill \end{array}& & & \begin{array}{ccc}& & \hfill 4+3\cdot 7\hfill \\ \text{Since }3\cdot 7\text{ is 21.}\hfill & & \hfill 4+21\hfill \\ \text{And }21+4\text{ makes 25.}\hfill & & \hfill 25\hfill \end{array}\hfill \end{array}[/latex]

Imagine the confusion that could result if every problem had several different correct answers. The same expression should give the same result. So mathematicians established some guidelines called the order of operations, which outlines the order in which parts of an expression must be simplified.

Order of Operations

When simplifying mathematical expressions perform the operations in the following order:
1. Parentheses and other Grouping Symbols

Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.

2. Exponents

Simplify all expressions with exponents.

3. Multiplication and Division

Perform all multiplication and division in order from left to right. These operations have equal priority.

4. Addition and Subtraction

Perform all addition and subtraction in order from left to right. These operations have equal priority.

Students often ask, “How will I remember the order?” Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase. Please Excuse My Dear Aunt Sally.

Order of Operations
Please	Parentheses
Excuse	Exponents
My Dear	Multiplication and Division
Aunt Sally	Addition and Subtraction

It’s good that ‘My Dear’ goes together, as this reminds us that multiplication and division have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.
Similarly, ‘Aunt Sally’ goes together and so reminds us that addition and subtraction also have equal priority and we do them in order from left to right.

example

Simplify the expressions:

[latex]4+3\cdot 7[/latex]
[latex]\left(4+3\right)\cdot 7[/latex]

Solution:

1.
	[latex]4+3\cdot 7[/latex]
Are there any parentheses? No.
Are there any exponents? No.
Is there any multiplication or division? Yes.
Multiply first.	[latex]4+\color{red}{3\cdot 7}[/latex]
Add.	[latex]4+21[/latex]
	[latex]25[/latex]

2.
	[latex](4+3)\cdot 7[/latex]
Are there any parentheses? Yes.	[latex]\color{red}{(4+3)}\cdot 7[/latex]
Simplify inside the parentheses.	[latex](7)7[/latex]
Are there any exponents? No.
Is there any multiplication or division? Yes.
Multiply.	[latex]49[/latex]

try it

example

Simplify:

[latex]\text{18}\div \text{9}\cdot \text{2}[/latex]
[latex]\text{18}\cdot \text{9}\div \text{2}[/latex]

Show Solution

try it

example

Simplify: [latex]18\div 6+4\left(5 - 2\right)[/latex].

Show Solution

try it

In the video below we show another example of how to use the order of operations to simplify a mathematical expression.

When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.

example

[latex]\text{Simplify: }5+{2}^{3}+3\left[6 - 3\left(4 - 2\right)\right][/latex].

Show Solution

try it

In the video below we show another example of how to use the order of operations to simplify an expression that contains exponents and grouping symbols.

example

Simplify: [latex]{2}^{3}+{3}^{4}\div 3-{5}^{2}[/latex].

Show Solution

	[latex]18\div 6+4(5-2)[/latex]
Parentheses? Yes, subtract first.	[latex]18\div 6+4(\color{red}{3})[/latex]
Exponents? No.
Multiplication or division? Yes.
Divide first because we multiply and divide left to right.	[latex]\color{red}{3}+4(3)[/latex]
Any other multiplication or division? Yes.
Multiply.	[latex]3+\color{red}{12}[/latex]
Any other multiplication or division? No.
Any addition or subtraction? Yes.	[latex]15[/latex]

	[latex]5+{2}^{3}+ 3[6-3(4-2)][/latex]
Are there any parentheses (or other grouping symbol)? Yes.
Focus on the parentheses that are inside the brackets.	[latex]5+{2}^{3}+ 3[6-3\color{red}{(4-2)}][/latex]
Subtract.	[latex]5+{2}^{3}+3[6-\color{red}{3(2)}][/latex]
Continue inside the brackets and multiply.	[latex]5+{2}^{3}+3[6-\color{red}{6}][/latex]
Continue inside the brackets and subtract.	[latex]5+{2}^{3}+3[\color{red}{0}][/latex]
The expression inside the brackets requires no further simplification.
Are there any exponents? Yes.
Simplify exponents.	[latex]5+\color{red}{{2}^{3}}+3[0][/latex]
Is there any multiplication or division? Yes.
Multiply.	[latex]5+8+\color{red}{3[0]}[/latex]
Is there any addition or subtraction? Yes.
Add.	[latex]\color{red}{5+8}+0[/latex]
Add.	[latex]\color{red}{13+0}[/latex]
	[latex]13[/latex]

	[latex]{2}^{3}+{3}^{4}\div 3-{5}^{2}[/latex]
If an expression has several exponents, they may be simplified in the same step.
Simplify exponents.	[latex]\color{red}{{2}^{3}}+\color{red}{{3}^{4}}\div 3-\color{red}{{5}^{2}}[/latex]
Divide.	[latex]8+\color{red}{81\div 3}-25[/latex]
Add.	[latex]\color{red}{8+27}-25[/latex]
Subtract.	[latex]\color{red}{35-25}[/latex]
	[latex]10[/latex]

Chapter 3: Introduction to Algebra