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Authors: Mark Zegarelli
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multiplication/division
9 + 6 ÷ 3
1. Evaluate multiplication and division left to right.2. Evaluate addition and subtraction left to right.
In this section, I show you how to identify and evaluate all three types of expressions.
Expressions with only addition and subtraction
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 â 5 + 3 â 8
Because the only operations are addition and subtraction, you can evaluate from left to right, starting with 17 â 5:
= 12 + 3 â 8
As you can see, the number 12 replaces 17 â 5. Now the expression has three numbers instead of four. Next, evaluate 12 + 3:
= 15 â 8
This step breaks down the expression to two numbers, which you can evaluate easily:
= 7
So 17 â 5 + 3 â 8 = 7.
Expressions with only multiplication and division
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:
Even though this expression has some negative numbers, the only operations it contains are multiplication and division. So you can evaluate it in two steps from left to right (remembering the rules for multiplying and dividing with negative numbers that I show you in Chapter 4 ):
Thus,.
Mixed-operator expressions
Often an expression contains
At least one addition or subtraction operator
At least one multiplication or division operator
I call these
mixed-operator expressions.
To evaluate them, you need some stronger medicine.
 Evaluate mixed-operator expressions as follows:
Evaluate the multiplication and division from left to right.
Evaluate the addition and subtraction from left to right.
For example, suppose you want to evaluate the following expression:
As you can see, this expression contains addition, multiplication, and division, so it's a mixed-operator expression. To evaluate it, start by underlining the multiplication and division in the expression:
Now evaluate what youâve underlined from left to right:
At this point, you're left with an expression that contains only addition, so you can evaluate it from left to right:
Thus,.
Using order of operations in expressions with exponents
Here's what you need to know to evaluate expressions that have exponents (see Chapter 4 for info on exponents).
 Evaluate exponents from left to right
before
you begin evaluating Big Four operations (adding, subtracting, multiplying, and dividing).
The trick here is to turn the expression into a Big Four expression and then use what I show you earlier in âApplying order of operations to Big Four expressions.â For example, suppose you want to evaluate the following:
First, evaluate the exponent:
At this point, the expression contains only addition and subtraction, so you can evaluate it from left to right in two steps:
So.
Understanding order of precedence in expressions with parentheses
In math, parentheses â ( ) â are often used to group together parts of an expression. When it comes to evaluating expressions, here's what you need to know about parentheses.
 To evaluate expressions that contain parentheses,
Evaluate the contents of parentheses from the inside out.
Evaluate the rest of the expression.
Big Four expressions with parentheses
Similarly, suppose you want to evaluate. This expression contains
9 + 6 ÷ 3
1. Evaluate multiplication and division left to right.2. Evaluate addition and subtraction left to right.
In this section, I show you how to identify and evaluate all three types of expressions.
Expressions with only addition and subtraction
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 â 5 + 3 â 8
Because the only operations are addition and subtraction, you can evaluate from left to right, starting with 17 â 5:
= 12 + 3 â 8
As you can see, the number 12 replaces 17 â 5. Now the expression has three numbers instead of four. Next, evaluate 12 + 3:
= 15 â 8
This step breaks down the expression to two numbers, which you can evaluate easily:
= 7
So 17 â 5 + 3 â 8 = 7.
Expressions with only multiplication and division
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:
Even though this expression has some negative numbers, the only operations it contains are multiplication and division. So you can evaluate it in two steps from left to right (remembering the rules for multiplying and dividing with negative numbers that I show you in Chapter 4 ):
Thus,.
Mixed-operator expressions
Often an expression contains
At least one addition or subtraction operator
At least one multiplication or division operator
I call these
mixed-operator expressions.
To evaluate them, you need some stronger medicine.
 Evaluate mixed-operator expressions as follows:
Evaluate the multiplication and division from left to right.
Evaluate the addition and subtraction from left to right.
For example, suppose you want to evaluate the following expression:
As you can see, this expression contains addition, multiplication, and division, so it's a mixed-operator expression. To evaluate it, start by underlining the multiplication and division in the expression:
Now evaluate what youâve underlined from left to right:
At this point, you're left with an expression that contains only addition, so you can evaluate it from left to right:
Thus,.
Using order of operations in expressions with exponents
Here's what you need to know to evaluate expressions that have exponents (see Chapter 4 for info on exponents).
 Evaluate exponents from left to right
before
you begin evaluating Big Four operations (adding, subtracting, multiplying, and dividing).
The trick here is to turn the expression into a Big Four expression and then use what I show you earlier in âApplying order of operations to Big Four expressions.â For example, suppose you want to evaluate the following:
First, evaluate the exponent:
At this point, the expression contains only addition and subtraction, so you can evaluate it from left to right in two steps:
So.
Understanding order of precedence in expressions with parentheses
In math, parentheses â ( ) â are often used to group together parts of an expression. When it comes to evaluating expressions, here's what you need to know about parentheses.
 To evaluate expressions that contain parentheses,
Evaluate the contents of parentheses from the inside out.
Evaluate the rest of the expression.
Big Four expressions with parentheses
Similarly, suppose you want to evaluate. This expression contains
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