3 + 4(5 − 1) − 32 × 2 = ?
Before you start dealing with variables, spend a moment looking at expressions that are comprised of only numbers, such as the preceding example. The GRE probably won’t ask you to compute something like this directly, but learning to use order of operations on numerical expressions will help you manipulate algebraic expressions and equations. So you have a string of numbers with mathematical symbols in between them. Which part of the expression should you focus on first?
Intuitively, most people think of going in the direction they read, from left to right. When you read a book, moving left to right is a wise move (unless you’re reading a language such as Chinese or Hebrew). However, when you perform basic arithmetic, there is an order that is of greater importance: the order of operations.
The order in which you perform the mathematical functions should primarily be determined by the functions themselves. In the correct order, the six operations are Parentheses, Exponents, Multiplication/Division, and Addition/Subtraction (or PEMDAS).
Before you solve a problem that requires PEMDAS, here’s a quick review of the basic operations:
Parentheses can be written as ( ) or [ ] or even { }.
Exponents are 52← these numbers. For example, 52 (“five squared”) can be expressed as 5 × 5. In other words, it is 5 times itself, or two 5’s multiplied together.
Likewise, 43 (“four cubed,” or “four to the third power”) can be expressed as 4 × 4 × 4 (or three 4’s multiplied together). The exponent 3 tells you how many 4’s are in the product.
Roots are very closely related to exponents. For example,
is the third root of 64 (commonly called the cube root). The cube root, in this case 
, is basically asking the question, “What multiplied by itself three times equals 64?” This is written as 4 × 4 × 4 = 64, so
. The plain old square root
can be thought of as
. “What times itself equals 9?” The answer is 3 × 3 = 9, so
.
Exponents and roots can also undo each other:
and
.
Multiplication and Division can also undo each other: 2 × 3 ÷ 3 = 2 and 10 ÷ 5 × 5 = 10.
Multiplication can be expressed with a multiplication sign (×) or with parentheses: (5)(4) = 5 × 4 = 20. Division can be expressed with a division sign (÷), a slash (/), or a fraction bar(—):
.
Also remember that multiplying or dividing by a negative number changes the sign:
| 4 × (−2) = −8 | −8 ÷ (−2) = 4 |
Addition and Subtraction can also undo each other: 8 + 7 − 7 = 8 and 15 − 6 + 6 = 15.
PEMDAS is a useful acronym you can use to remember the order in which operations should be performed. Some people find it useful to write PEMDAS like this:
For Multiplication/Division and Addition/Subtraction, perform whichever comes first from left to right. The reason that Multiplication and Division are at the same level of importance is that any Multiplication can be expressed as Division, and vice versa; for example, 7 ÷ 2 is equivalent to 7 ×
. In a sense, Multiplication and Division are two sides of the same coin.
Addition and Subtraction have this same relationship: 3 − 4 is equivalent to 3 + (−4).
The correct order of steps to simplify this sample expression is as follows:
| 3 + 4(5 − 1) − 32 × 2 | |
| Parentheses | 3 + 4(4) − 32 × 2 |
| Exponents | 3 + 4(4) − 9 × 2 |
| Multiplication or Division (left to right) | 3 + 16 − 18 |
| Addition or Subtraction (left to right) | 3 + 16 − 18 = 19 − 18 = 1 |
Remember: If you have two operations of equal importance, you should do them in left-to-right order: 3 − 2 + 3 = 1 + 3 = 4. The only instance in which you would override this order is when the operations are in parentheses: 3 − (2 + 3) = 3 − (5) = −2.
Next are two problems together. Try them first on your own, then an explanation will follow:
5 − 3 × 43 ÷ (7 − 1)
P
E
M/D
A/S
Your work should have looked like this:
Here’s one more:
32 ÷ 24 × (5 − 32)
P
E
M/D
A/S
Here’s the work you should have done:
Evaluate the following expressions.
−4 + 12/3 =
(5 − 8) × 10 − 7 =
−3 × 12 ÷ 4 × 8 + (4 − 6) =
24 × (8 ÷ 2 − 1)/(9 − 3) =