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.
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 in both the numerator and denominator at the same time.
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.
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 absolutel value, or distance from zero, to make sure the number is non-negative.
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.
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.
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. Let's take a look at an example:
Radical signs also act as a grouping symbol. If an expression is underneath a radical, the expression must be fully evaluated following the order of operations, before evaluating other parts of the expression.
Source: Adapted from "Beginning and Intermediate Algebra" by Tyler Wallace, an open source textbook available at: http://wallace.ccfaculty.org/book/book.html
An acronym to remember the order of operations: parentheses, exponents, multiplication and division, addition and subtraction.