Module 1.1 – Order of Operations

Learning Objectives

In this section, you will:

  • Review the order of operations.

Order of Operations

When we multiply a number by itself, we square it or raise it to a power of 2. For example, $latex 4^{2} = 4 \cdot 4 = 16$. We can raise any number to any power. In general, the exponential notation [latex]a^{n}[/latex] means that the number or variable [latex]a[/latex] is used as a factor [latex]n[/latex] times.

[latex]a^{n} = a \cdot a \cdot a \cdot \cdots \cdot a \quad\text{($n$ times)}[/latex]

In this notation, [latex]a^{n}[/latex] is read as the [latex]n[/latex]th power of [latex]a[/latex], where [latex]a[/latex] is called the base and [latex]n[/latex] is called the exponent. A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operations. For example, [latex]24 + 6 \cdot \frac{2}{3} - 4^{2}[/latex] is a mathematical expression.

To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the order of operations. This is a sequence of rules for evaluating such expressions.

Recall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols.

The next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right.

Let’s take a look at the expression provided.

[latex]24 + 6 \cdot \frac{2}{3} - 4^{2}[/latex]

There are no grouping symbols, so we move on to exponents or radicals. The number 4 is raised to a power of 2, so simplify [latex]4^{2}[/latex] as 16.

[latex]24 + 6 \cdot \frac{2}{3} - 4^{2} = 24 + 6 \cdot \frac{2}{3} - 16[/latex]

Next, perform multiplication or division, left to right.

[latex]24 + 6 \cdot \frac{2}{3} - 16 = 24 + 4 - 16[/latex]

Lastly, perform addition or subtraction, left to right.

[latex]24 + 4 - 16 = 28 - 16 = 12[/latex]

Therefore, [latex]24 + 6 \cdot \frac{2}{3} - 4^{2} = 12[/latex].

For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result.

order of operations
Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using
the acronym PEMDAS:
Parentheses (worked from innermost out)
Exponents
Multiplication and Division (worked left to right)
Addition and Subtraction (worked left to right)

Image of a partially green-shaded rectangle with length 10x and width 6x. Unshaded parts include top-left and top-right corner squares of length and width 4, and bottom middle rectangle of width 4 and unmarked length.How To…

Given a mathematical expression, simplify it using the order of operations.

  1. Simplify any expressions within grouping symbols.
  2. Simplify any expressions containing exponents or radicals.
  3. Perform any multiplication and division in order, from left to right.
  4. Perform any addition and subtraction in order, from left to right.

Example 1   Using the Order of Operations

Use the order of operations to evaluate each of the following expressions.

  1. [latex](3 \cdot 2)^{2} - 4(6 + 2)[/latex]
  2. [latex]\dfrac{5^{2} - 4}{7} - \sqrt{11-2}[/latex]
  3. [latex]6 - |5 - 8| + 3(4 - 1)[/latex]
  4. [latex]\dfrac{14 - 3 \cdot 2}{2 \cdot 5 - 3^{2}}[/latex]
  5. [latex]7(5 \cdot 3) - 2[(6 - 3) - 4^{2}] + 1[/latex]

Solution

  1. [latex]\quad[/latex]
    \begin{align*}
    (3 \cdot 2)^{2} – 4(6 + 2) &= (6)^{2} – 4(8)   &   \text{Simplify parentheses.} \\
    &= 36 – 4(8)   &   \text{Evaluate exponent.} \\
    &= 36 – 32   &   \text{Perform multiplication.} \\
    &= 4   &   \text{Perform subtraction.}
    \end{align*}
  2. Note that the fraction bar and radical here count as grouping symbols, and hence fall under the parentheses step. That is, we must simplify what is “inside” before evaluating the fraction or radical.
    \begin{align*}
    \dfrac{5^{2} – 4}{7} – \sqrt{11-2} &= \dfrac{25 – 4}{7} – \sqrt{11 – 2} \\
    &= \dfrac{21}{7} – \sqrt{9} \\
    &= \dfrac{21}{7} – 3 \\
    &= 3 – 3 \\
    &= 0
    \end{align*}
  3. [latex]\quad[/latex]
    \begin{align*}
    6 – |5 – 8| + 3(4 – 1) &= 6 – |-3| + 3(3) \\
    &= 6 – 3 + 3(3) \\
    &= 6 – 3 + 9 \\
    &= 3 + 9 \\
    &= 12
    \end{align*}
  4. As in part b. above, the fraction bar here acts as a grouping symbol, so we must simplify the numerator and denominator “groups” before evaluating the fraction.
    \begin{align*}
    \dfrac{14 – 3 \cdot 2}{2 \cdot 5 – 3^{2}} &= \dfrac{14 – 3 \cdot 2}{2 \cdot 5 – 9} \\
    &= \dfrac{14 – 6}{10 – 9} \\
    &= \dfrac{8}{1} \\
    &= 8
    \end{align*}
  5. [latex]\quad[/latex]
    \begin{align*}
    7(5 \cdot 3) – 2[(6 – 3) – 4^{2}] + 1 &= 7(15) – 2[(3) – 4^{2}] + 1 \\
    &= 7(15) – 2[3 – 16] + 1 \\
    &= 7(15) – 2[-13] + 1 \\
    &= 105 + 26 + 1 \\
    &= 132
    \end{align*}

Try It

Use the order of operations to evaluate each of the following expressions.

  1. [latex]\sqrt{5^{2} - 4^{2}} + 7(5-4)^{2}[/latex]
  2. [latex]1 + \dfrac{7 \cdot 5 - 8 \cdot 4}{9-6}[/latex]
  3. [latex]|1.8 - 4.3| + 0.4 \sqrt{15 + 10}[/latex]
  4. [latex]\frac{1}{2}[5 \cdot 3^{2} - 7^{2}] + \frac{1}{3} \cdot 9^{2}[/latex]
  5. [latex][(3 - 8)^{2} - 4] - (3 - 8)[/latex]

 

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Mathematics for Business Copyright © by Jonathan Brucks. All Rights Reserved.

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