Basic Math and Pre-Algebra For Dummies (21 page)

When you evaluate an arithmetic expression, you simplify it to a single numerical value — in other words, you find the number that it's equal to. For example, evaluate the following arithmetic expression:

How? Simplify it to a single number:

• 35
  

I'm sure you're dying to know how the Three E's — equations, expressions, and evaluation — are all connected.
Evaluation
allows you to take an
expression
containing more than one number and reduce it to a single number. Then you can make an
equation,
using an equals sign, to connect the expression and the number. For example, here's an
expression
containing four numbers:

• 1 + 2 + 3 + 4
  

When you
evaluate
it, you reduce it to a single number:

• 10
  

And now you can make an
equation
by connecting the expression and the number with an equals sign:

• 1 + 2 + 3 + 4 = 10
  

When you were a kid, did you ever try putting on your shoes first and then your socks? If you did, you probably discovered this simple rule:

1. Put on socks.
   
2. Put on shoes.
   

Thus, you have an order of operations: The socks have to go on your feet before your shoes. So in the act of putting on your shoes and socks, your socks have precedence over your shoes. A simple rule to follow, right?

In this section, I outline a similar set of rules for evaluating expressions, called the
order of operations
(sometimes called
order of precedence
). Don't let the long name throw you. Order of operations is just a set of rules to make sure you get your socks and shoes on in the right order, mathematically speaking, so you always get the right answer.

Note:
Through most of this book, I introduce overarching themes at the beginning of each section and then explain them later in the chapter instead of building them and finally revealing the result. But order of operations is a bit too confusing to present that way. Instead, I start with a list of four rules and go into more detail about them later in the chapter. Don't let the complexity of these rules scare you off before you work through them!

 Evaluate arithmetic expressions from left to right according to the following order of operations:

1. Parentheses
   
2. Exponents
   
3. Multiplication and division
   
4. Addition and subtraction
   

Don't worry about memorizing this list right now. I break it to you slowly in the remaining sections of this chapter, starting from the bottom and working toward the top, as follows:

• In “Applying order of operations to Big Four expressions,” I show Steps 3 and 4 — how to evaluate expressions with any combination of addition, subtraction, multiplication, and division.
  
• In “Using order of operations in expressions with exponents,” I show you how Step 2 fits in — how to evaluate expressions with Big Four operations plus exponents, square roots, and absolute value.
  
• In “Understanding order of operations in expressions with parentheses,” I show you how Step 1 fits in — how to evaluate all the expressions I explain plus expressions with parentheses.
  

As I explain earlier in this chapter, evaluating an expression is just simplifying it to a single number. Now I get you started on the basics of evaluating expressions that contain any combination of the Big Four operations — adding, subtracting, multiplying, and dividing. (For more on the Big Four, see Chapter
3
.) Generally speaking, the Big Four expressions come in the three types in Table 
5-1
.

Table 5-1 The Three Types of Big Four Expressions

In this section, I show you how to identify and evaluate all three types of expressions.

Some expressions contain only addition and subtraction. When this is the case, the rule for evaluating the expression is simple.

 When an expression contains only addition and subtraction, evaluate it step by step from left to right. For example, suppose you want to evaluate this expression:

• 17 − 5 + 3 − 8
  

Because the only operations are addition and subtraction, you can evaluate from left to right, starting with 17 − 5:

• = 12 + 3 − 8
  

As you can see, the number 12 replaces 17 − 5. Now the expression has three numbers instead of four. Next, evaluate 12 + 3:

• = 15 − 8
  

This step breaks down the expression to two numbers, which you can evaluate easily:

• = 7
  

So 17 − 5 + 3 − 8 = 7.

Some expressions contain only multiplication and division. When this is the case, the rule for evaluating the expression is pretty straightforward.

 When an expression contains only multiplication and division, evaluate it step by step from left to right. Suppose you want to evaluate this expression:

Again, the expression contains only multiplication and division, so you can move from left to right, starting with
:

Notice that the expression shrinks one number at a time until all that's left is 2. So
.

Here's another quick example: