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Operations as Grouping Symbols

Author: Sophia
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what's covered
In this lesson, you will learn how to evaluate an expression with absolute value using order of operations. Specifically, this lesson will cover:

Table of Contents

1. Operations as Grouping Symbols

The order of operations outlines an order we should follow to evaluate expressions with multiple operations. We often use the acronym PEMDAS to remember the order of operations:

It is important to remember that the "parentheses" part of the order of operations includes grouping symbols, such as braces or brackets. There are several types of operations that act as grouping symbols. This means that they are operations, but need to be considered first in the order of operations, because they also enclose other expressions in parentheses, even though they might not be visible.

In this lesson, we will see three operations that act as grouping symbols: fraction bars, absolute value bars, and the radical symbol.

term to know
PEMDAS
An acronym to remember the order of operations: parentheses, exponents, multiplication and division (from left to right), addition and subtraction (from left to right).

1a. Fraction Bars

If we have a fraction, the entire numerator and the entire denominator must be evaluated before we reduce the fraction. In these cases, we can simplify both the numerator and denominator at the same time.

EXAMPLE

fraction numerator 2 to the power of 4 minus open parentheses short dash 8 close parentheses times 3 over denominator stack 15 divided by 5 with underbrace below minus 1 end fraction Exponent in the numerator, divide in denominator
fraction numerator 16 minus stack open parentheses short dash 8 close parentheses times 3 with overbrace on top over denominator stack 3 minus 1 with underbrace below end fraction Multiply in the numerator, subtract in the denominator
fraction numerator stack 16 minus open parentheses short dash 24 close parentheses with overbrace on top over denominator 2 end fraction Add the opposite to simplify numerator, denominator is done
40 over 2 Reduce, divide
20 Our Solution

big idea
A fraction bar groups numbers and operations in a numerator and denominator. Although a fraction bar represents division, it creates implied parentheses around everything on top and on the bottom. We must fully evaluate both the numerator and denominator of a fraction, and divide at the very end.

1b. Absolute Value Bars

Another type of grouping symbol that also has an operation with it is absolute value. When we have an absolute value, we will evaluate everything inside the absolute value bars, just as if it were a normal parentheses. Then once the inside is completed, we will take the absolute value, or distance from zero, to make sure the number is non-negative.

EXAMPLE

1 plus 3 open vertical bar short dash left parenthesis 4 squared right parenthesis minus open parentheses short dash 8 close parentheses close vertical bar plus 2 open vertical bar 3 plus stack open parentheses short dash 5 close parentheses squared with underbrace below close vertical bar Evaluate absolute first, exponents
1 plus 3 open vertical bar stack short dash 16 minus open parentheses short dash 8 close parentheses with underbrace below close vertical bar plus 2 open vertical bar stack 3 plus 25 with underbrace below close vertical bar Add inside absolute values
1 plus 3 open vertical bar stack short dash 8 with underbrace below close vertical bar plus 2 open vertical bar 28 with underbrace below close vertical bar Evaluate absolute values
1 plus stack 3 open parentheses 8 close parentheses with underbrace below plus 2 open parentheses 28 close parentheses Multiply left to right
1 plus 24 plus stack 2 open parentheses 28 close parentheses with underbrace below Finish multiplying
stack 1 plus 24 with underbrace below plus 56 Add left to right
stack 25 plus 56 with underbrace below Add
81 Our Solution

big idea
Absolute value bars also act as a grouping symbol, and come first in the order of operations. If we see an expression inside absolute value bars, we must completely evaluate that expression, then take its absolute value, before moving on to the other operations according to the order of operations.

hint
The above example also illustrates an important point about exponents. Exponents are considered to be only on the number they are attached to. This means when we see -42, only the 4 is squared, giving us -(42) or -16. But when the negative is in parentheses, such as (-5)2, the negative is part of the number and is also squared, giving us a positive solution, 25.

1c. Radical Symbols

Expressions can be underneath radical symbols, just like they can be in numerators, denominators, or within absolute value bars. When an expression is underneath a radical, this groups the expression together, and acts as a grouping symbol. This means we must fully evaluate the expression underneath the radical sign, before moving on to other operations in the order of operations.

EXAMPLE

square root of 4 times 5 plus 5 end root plus 7 The radical sign groups the expression 4 times 5 plus 5
square root of 20 plus 5 end root plus 7 Multiply 4 by 5
square root of 25 plus 7 Add 5
5 plus 7 Evaluate the square root
12 Our Solution

summary
Grouping symbols fall under parentheses in PEMDAS, which means that when you're evaluating with the order of operations, you're going to start with grouping symbols first. Grouping symbols include fraction bars, absolute value signs, and radicals. For nested grouping symbols, where you have more than one grouping symbol inside of another one, you need to start by evaluating them from the inside out. Whatever your innermost grouping symbol is, start with that, and then work your way out.

Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License

Terms to Know
PEMDAS

An acronym to remember the order of operations: parentheses, exponents, multiplication and division, addition and subtraction. 

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