Simplify using the order of operations agreement

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]\begin\hfill \text\hfill & & & \hfill \text\hfill \\ \begin& & \hfill 4+3\cdot 7\hfill \\ \text4+3\text< gives 7.>\hfill & & \hfill 7\cdot 7\hfill \\ \text7\cdot 7\text< is 49.>\hfill & & \hfill 49\hfill \end& & & \begin& & \hfill 4+3\cdot 7\hfill \\ \text3\cdot 7\text< is 21.>\hfill & & \hfill 4+21\hfill \\ \text21+4\text< makes 25.>\hfill & & \hfill 25\hfill \end\hfill \end[/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]

The table shows the expression four plus three times seven. On the next line it states are there any parentheses in the expression? No. The line below that states are there any exponents in the expression? No. The next line states is there any multiplication or division in the expression? Yes. The next line states Multiply first and is followed by the expression of four plus three times seven. The expression is now four plus twenty-one. The last operation is addition. Add four and twenty-one to get twenty-five.">

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[/latex]
Add.	[latex]4+21[/latex]
[latex]25[/latex]

four and three to get seven. The expression is now seven times seven. Is there any multiplication or division? Yes, multiply seven by seven to get forty-nine.">

2.
[latex](4+3)\cdot 7[/latex]
Are there any parentheses? Yes.	[latex]\color\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]