Using the Order of Operations

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

$\text{\textcircled{a}} \enspace {3\cdot2}^2-4(6+2)$ $\quad$ $\text{\textcircled{b}} \enspace \frac{5^2-4}{7}-\sqrt{11-2}$ $\quad$ $\text{\textcircled{c}} \enspace 6-|5-8|+3(4-1)$ $\quad$ $\text{\textcircled{d}} \enspace \frac{14-3\cdot2}{2\cdot5-3^2}$ $\quad$ $\text{\textcircled{e}} \enspace 7(5\cdot3)-2|(6-3)-4^2|+1$

Solution

$\text{\textcircled{a}}$

$$\begin{alignat*}{3} {3\cdot2}^2-4(6+2) &= (6)^2-4(8) &&\quad \text{Simplify parentheses.} \\ &= 36-4(8) &&\quad \text{Simplify exponent.} \\ &= 36-32 &&\quad \text{Simplify multiplication.} \\ &= 4 &&\quad \text{Simplify multiplication.} \\ \end{alignat*}$$

$\text{\textcircled{b}}$

$$\begin{alignat*}{3} \frac{5^2-4}{7}-\sqrt{11-2} &= \frac{5^2-4}{7}-\sqrt{9} &&\quad \text{Simplify grouping symbols (radical).} \\ &= \frac{5^2-4}{7}-3 &&\quad \text{Simplify radical.} \\ &= \frac{25-4}{7}-3 &&\quad \text{Simplify exponent.} \\ &= \frac{21}{7}-3 &&\quad \text{Simplify subtraction in numerator.} \\ &= 3-3 &&\quad \text{Simplify division.} \\ &= 0 &&\quad \text{Simplify subtraction.} \\ \end{alignat*}$$

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.

$\text{\textcircled{c}}$

$$\begin{alignat*}{3} 6-|5-8|+3(4-1) &= 6-|-3|+3(3) &&\quad \text{Simplify inside grouping symbols.} \\ &= 6-3+3(3) &&\quad \text{Simplify absolute value.} \\ &= 6-3+9 &&\quad \text{Simplify multiplcation.} \\ &= 3+9 &&\quad \text{Simplify subtraction.} \\ &= 12 &&\quad \text{Simplify addition.} \\ \end{alignat*}$$

$\text{\textcircled{d}}$

$$\begin{alignat*}{3} \frac{14-3\cdot2}{2\cdot5-3^2} &= \frac{14-3\cdot2}{2\cdot5-9} &&\quad \text{Simplify exponent.} \\ &= \frac{14-6}{10-9} &&\quad \text{Simplify products.} \\ &= \frac{8}{1} &&\quad \text{Simplify differences.} \\ &= 8 &&\quad \text{Simplify quotient.} \\ \end{alignat*}$$

In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.

$\text{\textcircled{e}}$

$$\begin{alignat*}{3} 7(5\cdot3)-2|(6-3)-4^2|+1 &= 7(15)-2|(3)-4^2|+1 &&\quad \text{Simplify inside parentheses.} \\ &= 7(15)-2(3-16)+1 &&\quad \text{Simplify absolute value.} \\ &= 7(15)-2(-13)+1 &&\quad \text{Subtract.} \\ &= 105+26+1 &&\quad \text{Multiply.} \\ &= 132 &&\quad \text{Add.} \\ \end{alignat*}$$

Case

The above case, including the title and solution, is attributed to College Algebra 2e. Page 13-14.

Q&A

What are order of operations?

Page 12-13, College Algebra 2e

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)

Glossary

Denominator

https://en.wiktionary.org/wiki/denominator

The number or expression written below the line in a fraction (such as 2 in ½).

Last accessed

December 26, 2023

Fraction

https://en.wikipedia.org/wiki/Fraction

A fraction represents a part of a whole or, more generally, any number of equal parts.

Last accessed

December 26, 2023

Numerator

https://en.wiktionary.org/wiki/numerator

The number or expression written above the line in a fraction (such as 1 in ½).

Last accessed

December 26, 2023

Order of Operations

https://en.wikipedia.org/wiki/Order_of_operations

The order of operations is a collection of rules that reflect conventions about which operations to perform first in order to evaluate a given mathematical expression.

Last accessed

January 01, 2024

note

• Content last updated on January 01, 2024
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