Numbers and Their Applications: Background You’ll Need 2

  • Use PEMDAS

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 order when you are required to use several mathematical operations for one expression.

How to: Perform the Order of Operations

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

This order of operations is true for all real numbers.

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 [latex]\displaystyle \Rightarrow[/latex] Parentheses (and other grouping symbols)
  • Excuse [latex]\displaystyle \Rightarrow[/latex] Exponents
  • My Dear [latex]\displaystyle \Rightarrow[/latex] Multiplication and Division (from left to right)
  • Aunt Sally [latex]\displaystyle \Rightarrow[/latex] Addition and Subtraction (from left to right)

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 you about this!

Simplify the following:

[latex]7–5+3\cdot8[/latex]

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.

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], [latex]7[/latex]is the base and [latex]2[/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.

Simplify the following:

[latex]3^{2}\cdot2^{3}[/latex]

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

Simplify the following:

 [latex]\Large\frac{\sqrt{7+2}+2^2}{(8)(4)-11}[/latex]