Performing Calculations Using the Order of Operations

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

a^{n} = a \cdot a \cdot a \cdot \dots \cdot a

$a^{n}=a\cdot a\cdot a\cdot \dots \cdot a$

In this notation, ${a}^{n}$ is read as the nth power of $a$ , where $a$ is called the base and $n$ 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, $24+6\cdot \frac{2}{3}-{4}^{2}$ 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.

24 + 6 \cdot \frac{2}{3} - 4^{2}

$24+6\cdot \frac{2}{3}-{4}^{2}$

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 ${4}^{2}$ as 16.

\begin{matrix} 4 + 6 \cdot \frac{2}{3} - 4^{2} \\ 24 + 6 \cdot 23 - 16 \end{matrix}

$\begin{array}24+6\cdot\frac{2}{3}-4^{2} \\ 24+6\cdot{2}{3}-16\end{array}$

Next, perform multiplication or division, left to right.

\begin{matrix} 4 + 6 \cdot \frac{2}{3} - 16 \\ 24 + 4 - 16 \end{matrix}

$\begin{array}24+6\cdot\frac{2}{3}-16 \\ 24+4-16\end{array}$

Lastly, perform addition or subtraction, left to right.

\begin{matrix} 4 + 4 - 16 \\ 28 - 16 \\ 12 \end{matrix}

$\begin{array}24+4-16 \\ 28-16 \\ 12\end{array}$

Therefore, $24+6\cdot \frac{2}{3}-{4}^{2}=12$ .

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.

A General Note: Order of Operations

Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym PEMDAS:

P(arentheses)

E(xponents)

M(ultiplication) and D(ivision)

A(ddition) and S(ubtraction)

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

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

Example 6: Using the Order of Operations

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

${\left(3\cdot 2\right)}^{2}-4\left(6+2\right)$
$\frac{{5}^{2}-4}{7}-\sqrt{11 - 2}$
$6-|5 - 8|+3\left(4 - 1\right)$
$\frac{14 - 3\cdot 2}{2\cdot 5-{3}^{2}}$
$7\left(5\cdot 3\right)-2\left[\left(6 - 3\right)-{4}^{2}\right]+1$

Solution

$\begin{array}{cccc}\left(3\cdot 2\right)^{2} \hfill& =\left(6\right)^{2}-4\left(8\right) \hfill& \text{Simplify parentheses} \\ \hfill& =36-4\left(8\right) \hfill& \text{Simplify exponent} \\ \hfill& =36-32 \hfill& \text{Simplify multiplication} \\ \hfill& =4 \hfill& \text{Simplify subtraction}\end{array}$
$\begin{array}{cccc}\frac{5^{2}}{7}-\sqrt{11-2} \hfill& =\frac{5^{2}-4}{7}-\sqrt{9} \hfill& \text{Simplify grouping systems (radical)} \\ \hfill& =\frac{5^{2}-4}{7}-3 \hfill& \text{Simplify radical} \\ \hfill& =\frac{25-4}{7}-3 \hfill& \text{Simplify exponent} \\ \hfill& =\frac{21}{7}-3 \hfill& \text{Simplify subtraction in numerator} \\ \hfill& =3-3 \hfill& \text{Simplify division} \\ \hfill& =0 \hfill& \text{Simplify subtraction}\end{array}$

Note that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol so the numerator is considered to be grouped.
$\begin{array}{cccc}6-|5-8|+3\left(4-1\right) \hfill& =6-|-3|+3\left(3\right) \hfill& \text{Simplify inside grouping system} \\ \hfill& =6-3+3\left(3\right) \hfill& \text{Simplify absolute value} \\ \hfill& =6-3+9 \hfill& \text{Simplify multiplication} \\ \hfill& =3+9 \hfill& \text{Simplify subtraction} \\ \hfill& =12 \hfill& \text{Simplify addition}\end{array}$
$\begin{array}{cccc}\frac{14-3\cdot2}{2\cdot5-3^{2}} \hfill& =\frac{14-3\cdot2}{2\cdot5-9} \hfill& \text{Simplify exponent} \\ \hfill& =\frac{14-6}{10-9} \hfill& \text{Simplify products} \\ \hfill& =\frac{8}{1} \hfill& \text{Simplify quotient} \\ \hfill& =8 \hfill& \text{Simplify quotient}\end{array}$
In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.
$\begin{array}{cccc}7\left(5\cdot3\right)-2[\left(6-3\right)-4^{2}]+1 \hfill& =7\left(15\right)-2[\left(3\right)-4^{2}]+1 \hfill& \text{Simplify inside parentheses} \\ \hfill& 7\left(15\right)-2\left(3-16\right)+1 \hfill& \text{Simplify exponent} \\ \hfill& =7\left(15\right)-2\left(-13\right)+1 \hfill& \text{Subtract} \\ \hfill& =105+26+1 \hfill& \text{Multiply} \\ \hfill& =132 \hfill& \text{Add}\end{array}$

Try It 6

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

a. $\sqrt{{5}^{2}-{4}^{2}}+7{\left(5 - 4\right)}^{2}$
b. $1+\frac{7\cdot 5 - 8\cdot 4}{9 - 6}$
c. $|1.8 - 4.3|+0.4\sqrt{15+10}$
d. $\frac{1}{2}\left[5\cdot {3}^{2}-{7}^{2}\right]+\frac{1}{3}\cdot {9}^{2}$
e. $[{\left(3 - 8\right)}^{2}-4]-\left(3 - 8\right)$

Solution

Real Numbers: Algebra Essentials