3.3 Order of Operations

A close-up view of a notebook show a mathematical calculation. A calculator and a pen are placed by the notebook. — **Figure 3.19** Calculators may automatically apply order of operations to calculations.Calculators may automatically apply order of operations to calculations. (credit: “Precision” by Leonid Mamchenkov/Flickr, CC BY 2.0)

After completing this module, you should be able to:

Simplify expressions using order of operations.
Simplify expressions using order of operations involving grouping symbols.

Calculates else sure someone be rules expect explicit we what that needs to need make, we to that them what be calculate to calculated first.

You probably read that sentence and couldn't make heads or tails of it. Seems like it might concern calculations, but maybe it concerns needs? You may even be attempting to unscramble the sentence as you read it, placing words in the order you might expect them to appear in. The reason that the sentence makes no sense is that the words don't follow the order you expect them to follow. Unscrambled, the sentence was intended to be “To be sure that someone else calculates when we expect them to calculate, we need rules that make explicit what needs to be calculated first.”

Similarly, when working with math expressions and equations, if we don't follow the rules for order of operations, arithmetic expressions make no sense. Just a simple expression would be problematic if we didn't have some rules to tell us what to calculate first. For instance, $4 \times 2^{2} + 3 + 5^{2}$ can be calculated in many ways. You could get 5,184. Or, you could get 80. Or, 96. The issue is that without following a set of rules for calculation, the same expression will give various results. In case you are curious, using the appropriate order of operations, we find $4 \times 2^{2} + 3 + 5^{2} = 44$ .

Simplify Expressions Using Order of Operations

The order in which mathematical operations is performed is a convention that makes it easier for anyone to correctly calculate. They follow the acronym EMDAS:

E	Exponents
M/D	Multiplication and division
A/S	Addition and Subtraction

So, what does EMDAS tell us to do? In an equation, moving left to right, we begin by calculating all the exponents first. Once the exponents have been calculated, we again move left to right, calculating the multiplications and divisions, one at a time. Multiplication and division hold the same position in the ordering, so when you encounter one or the other at this step, do it. Once the multiplications and divisions have been calculated, we again move left to right, calculating the additions and subtractions, one at a time. Additions and subtractions hold the same position in the ordering, so when you encounter one or the other at this step, do it. (You may have previously learned the order of operations as PEMDAS, with parentheses first; we will add that aspect later on.) We’ll explore this as we work an example.

Even if the expression being calculated gets more complicated, we perform the operations in the order: EMDAS.

Example 4 Using Eight Order of Operations

Correctly apply the order of operations to compute the following:

$4 - 25 \times 6 / 10 \times 3^{2} + 7 \times 2^{3}$ .

Step 1: To do so, calculate the exponents first, moving left to right. There are two occurrences of exponents in the expression, 3 squared and 2 cubed.

$4 - 25 \times 6 / 10 \times 3^{2} + 7 \times 2^{3} = 4 - 25 \times 6 / 10 \times 9 + 7 \times 8$

Step 2: Now that the exponents are calculated, perform the multiplication and division, moving left to right. The first is the product of 25 and 6.

$\begin{array}{l} 4 - 25 \times 6 / 10 \times 9 + 7 \times 8 \\ = 4 - 150 / 10 \times 9 + 7 \times 8 \end{array}$

Step 3: Next is the 150 divided by 10.

$\begin{array}{l} 4 - 150 / 10 \times 9 + 7 \times 8 \\ = 4 - 15 \times 9 + 7 \times 8 \end{array}$

Step 4: Next is 15 multiplied by 9.

$\begin{array}{l} 4 - 15 \times 9 + 7 \times 8 \\ = 4 - 135 + 7 \times 8 \end{array}$

Step 5: Finally, multiply the 7 and 8.

$\begin{array}{l} 4 - 135 + 7 \times 8 \\ = 4 - 135 + 56 \end{array}$

As all the multiplications and divisions have been calculated, the additions and subtractions are performed, moving left to right.

$\begin{array}{l} 4 - 135 + 56 \\ = - 131 + 56 \\ = - 131 + 56 \\ = - 75 \end{array}$

The computed value is −75.

Using the Order of Operations Involving Grouping Symbols

We have examined how to use the order of operations, denoted by EMDAS, to correctly calculate expressions. However, there may be expressions where a multiplication should happen before an exponent, or a subtraction before a division. To indicate an operation should be performed out of order, the operation is placed inside parentheses. When parentheses are present, the operations inside the parentheses are performed first. Adding the parentheses to our list, we now have PEMDAS, as shown below.

P	Parentheses
E	Exponents
M/D	Multiplication and division	(division is just the multiplication by the reciprocal)
A/S	Addition and subtraction	(subtraction is just the addition of the negative)

As said previously, parentheses indicate that some operation or operations will be performed outside the standard order of operation rules. For instance, perhaps you want to multiply 4 and 7 before squaring. To indicate that the multiplication happens before the exponent, the multiplication is placed inside parentheses: $open parenthesis 4 times 7 close parenthesis squared$ .

This means operations inside the parentheses take precedence, or happen before other operations. Now, the first step in calculating arithmetic expressions using the order of operations is to perform operations inside parentheses first. Inside the parentheses, you follow the order of operation rules EMDAS.

Be aware that there can be more than one set of parentheses, and parentheses within parentheses. When one set of parentheses is inside another set, do the innermost set first, and then work outward.

Example 8 Working Innermost Parentheses in the Order of Operations

Correctly apply the rules for order of operations to accurately compute the following:

$4 + 2 \times (3^{2} - {(2 + 5)}^{2} \times 4) / (3 + 8)$ .

Step 1: Perform all calculations within the parentheses before other operations. Evaluate the innermost parentheses first. We can work separate parentheses expressions at the same time. The innermost set of parentheses has the 2 + 5 inside. The 3 + 8 is in a separate set of parentheses, so that addition can occur at the same time as the 2 + 5.

$\begin{array}{l} 4 + 2 \times (3^{2} - {(2 + 5)}^{2} \times 4) / (3 + 8) \\ = 4 + 2 \times (3^{2} - {(7)}^{2} \times 4) / (11) \end{array}$

Step 2: Now that those parentheses have been handled, move on to the next set of parentheses. Applying the order of operation rules inside that set of parentheses, the exponent is evaluated first, then the multiplication, and then the addition.

$\begin{array}{l} 4 + 2 \times (3^{2} - 7^{2} \times 4) / 11 \\ = 4 + 2 \times (3^{2} - 49 \times 4) / 11 \\ = 4 + 2 \times (- 187) / 11 \end{array}$

Step 3: Since all parentheses have been cleared, apply the EMDAS rules to finish the calculation.

$4 + 2 \times (- 187) / 11 = 4 - 374 / 11 = 4 - 34 = - 30$

Key Terms

order of operations
PEMDAS

Key Concepts

Establishing shared rules on which arithmetic operations are calculated first is necessary. Without them, different people may find different values for the same expression.
The highest precedence is with expressions in parentheses. This allows parts of an expression to be calculated in an order different than the basic order of operations.
The lowest precedence is addition and subtraction, as they are the basis for all other calculations.
Multiplication and division have precedence over addition and subtraction, as they are representations of repeated addition or subtraction.
Exponents have precedence over multiplication and division, as they represent repeated multiplication and division.

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Adapted from Contemporary Mathematics by OpenStax (openstax.org), licensed under CC BY-NC-SA 4.0. Changes were made. License: CC-BY-NC-SA-4.0.