## Easy Mathematics Step-by-Step (2012)

### Chapter 4. Order of Operations

In this chapter, you apply your skills in computation to perform a series of indicated numerical operations. This chapter lays the foundation for numerical calculations by introducing you to the order of operations.

**Grouping Symbols**

Grouping symbols such as parentheses ( ), brackets [ ], and braces { } are used to keep things together that belong together. Fraction bars and absolute value bars are grouping symbols as well. When you do computations in numerical expressions, do operations in grouping symbols first. It is *very important* that you do so when you have addition or subtraction inside the grouping symbol.

Keep in mind, however, that parentheses also are used to indicate multiplication as in or for clarity as in .

Fraction bars indicate division. For instance, means .

Grouping symbols say “Do me first!”

**Problem** Evaluate each expression.

**Solution**

*Step 1*. Parentheses are a grouping symbol, so do 1 + 1 *first*.

When you no longer need the grouping symbol, omit it.

*Step 2*. Evaluate 2^{4}.

*Step 3*. Review the main results.

; , but . Not performing the addition, , inside the parentheses *first* can lead to an incorrect result.

*Step 1*. The fraction bar is a grouping symbol, so do the addition, , over the fraction bar *first*.

*Step 2*. Simplify .

. Not performing the addition, , *first* can lead to an incorrect result.

*Step 3*. Review the main results.

*Step 1*. The fraction bar is a grouping symbol, so do the addition, , over the fraction bar and the subtraction, , under the fraction bar *first*.

*Step 2*. Compute .

; , but . Not performing the addition, , and the subtraction, , *first* can result in a cancellation error.

*Step 3*. Review the main results.

*Step 1*. Absolute value bars are a grouping symbol, so do *first*.

*Step 2*. Evaluate .

*Step 3*. Review the main results.

; , but . Not performing the addition, , *first* can lead to an incorrect result.

**PEMDAS**

Mathematical expressions like and are *numerical expressions*. When you evaluate numerical expressions by doing the indicated math, you must follow the order of operations. Use the mnemonic “**P**lease **E**xcuse **M**y **D**ear **A**unt **S**ally”—abbreviated as PE(MD)(AS)—to help you remember the following order.

**Order of Operations**

1. Do computations inside **P**arentheses (or other grouping symbols).

2. Do **E**xponents (also, evaluate absolute values and roots).

3. Do **M**ultiplication and **D**ivision, in the order in which they occur from left to right.

4. Do **A**ddition and **S**ubtraction, in the order in which they occur from left to right.

In the order of operations, multiplication does not always have to be done before division, or addition before subtraction. You multiply and divide in the order in which they occur in the problem. Similarly, you add and subtract in the order in which they occur in the problem.

**Problem** Evaluate the expression.

**Solution**

*Step 1*. Do inside the parentheses.

*Step 2*. Do the exponent: 2^{3}.

*Step 3*. Do multiplication and division, from left to right.

Do because division occurs first.

. Multiply *before* adding or subtracting—when no grouping symbols are present.

*Step 4*. Do addition and subtraction, from left to right.

Do because subtraction occurs first.

*Step 5*. Review the main results.

*Step 1*. Compute inside the parentheses.

*Step 2*. Do the exponent: 3^{2}.

; , but . Do exponents *before* multiplication.

*Step 3*. Do multiplication and division, from left to right.

Do because multiplication occurs first.

. Do multiplication *before* addition (except when a grouping symbol indicates otherwise).

. Do division *before* subtraction (except when a grouping symbol indicates otherwise).

*Step 4*. Do addition and subtraction, from left to right.

Do because addition occurs first.

*Step 5*. Review the main results.

*Step 1*. Compute quantities in grouping symbols.

*Step 2*. Evaluate and 2^{3}.

Evaluate absolute value expressions and exponents *before* multiplication or division.

*Step 3*. Do division: .

*Step 4*. Do addition and subtraction, from left to right.

Do because addition occurs first.

*Step 5*. Review the main results.

*Step 1*. Compute quantities in parentheses.

*Step 2*. Do subtraction: .

*Step 3*. Review the main results.

*Step 1*. Do the exponent: (–1)^{2}.

*Step 2*. Do multiplication: 2·1.

. You can use a raised dot to show multiplication.

*Step 3*. Do addition.

*Step 4*. Review the main results.

*Step 1*. Do multiplication: .

*Step 2*. Do addition.

*Step 3*. Review the main results.

; , but . Don’t add (or subtract) before multiplying, unless you have a grouping symbol that tells you to do so.

*Step 1*. Do 2×5 inside the parentheses.

*Step 2*. Do the exponents.

*Step 3*. Do multiplication: .

; , but .

*Step 4*. Do addition.

*Step 5*. Review the main results.

because parentheses and multiplication come before addition.

*Step 1*. Do parentheses.

*Step 2*. Do multiplication.

*Step 3*. Do addition.

*Step 4*. Review the main results.

. You can use parentheses to show multiplication.

**Exercise 4**

*Evaluate the expression*.