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, [latex]{4}^{2}=4\cdot 4=16[/latex]. 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.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. [latex]{\left(3\cdot 2\right)}^{2}4\left(6+2\right)[/latex]
 [latex]\frac{{5}^{2}4}{7}\sqrt{11  2}[/latex]
 [latex]65  8+3\left(4  1\right)[/latex]
 [latex]\frac{14  3\cdot 2}{2\cdot 5{3}^{2}}[/latex]
 [latex]7\left(5\cdot 3\right)2\left[\left(6  3\right){4}^{2}\right]+1[/latex]
Solution

[latex]\begin{array}{cccc}\left(3\cdot 2\right)^{2} \hfill& =\left(6\right)^{2}4\left(8\right) \hfill& \text{Simplify parentheses} \\ \hfill& =364\left(8\right) \hfill& \text{Simplify exponent} \\ \hfill& =3632 \hfill& \text{Simplify multiplication} \\ \hfill& =4 \hfill& \text{Simplify subtraction}\end{array}[/latex]

[latex]\begin{array}{cccc}\frac{5^{2}}{7}\sqrt{112} \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{254}{7}3 \hfill& \text{Simplify exponent} \\ \hfill& =\frac{21}{7}3 \hfill& \text{Simplify subtraction in numerator} \\ \hfill& =33 \hfill& \text{Simplify division} \\ \hfill& =0 \hfill& \text{Simplify subtraction}\end{array}[/latex]
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.  [latex]\begin{array}{cccc}658+3\left(41\right) \hfill& =63+3\left(3\right) \hfill& \text{Simplify inside grouping system} \\ \hfill& =63+3\left(3\right) \hfill& \text{Simplify absolute value} \\ \hfill& =63+9 \hfill& \text{Simplify multiplication} \\ \hfill& =3+9 \hfill& \text{Simplify subtraction} \\ \hfill& =12 \hfill& \text{Simplify addition}\end{array}[/latex]
 [latex]\begin{array}{cccc}\frac{143\cdot2}{2\cdot53^{2}} \hfill& =\frac{143\cdot2}{2\cdot59} \hfill& \text{Simplify exponent} \\ \hfill& =\frac{146}{109} \hfill& \text{Simplify products} \\ \hfill& =\frac{8}{1} \hfill& \text{Simplify quotient} \\ \hfill& =8 \hfill& \text{Simplify quotient}\end{array}[/latex] In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.
 [latex]\begin{array}{cccc}7\left(5\cdot3\right)2[\left(63\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(316\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}[/latex]
Try It 6
Use the order of operations to evaluate each of the following expressions.a. [latex]\sqrt{{5}^{2}{4}^{2}}+7{\left(5  4\right)}^{2}[/latex] b. [latex]1+\frac{7\cdot 5  8\cdot 4}{9  6}[/latex] c. [latex]1.8  4.3+0.4\sqrt{15+10}[/latex] d. [latex]\frac{1}{2}\left[5\cdot {3}^{2}{7}^{2}\right]+\frac{1}{3}\cdot {9}^{2}[/latex] e. [latex][{\left(3  8\right)}^{2}4]\left(3  8\right)[/latex]
SolutionLicenses & Attributions
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 College Algebra. Provided by: OpenStax Authored by: OpenStax College Algebra. Located at: https://cnx.org/contents/[email protected]:1/Preface. License: CC BY: Attribution.