Order of operations is the correct sequence for performing mathematical calculations. The order of operations is important because it can affect the result of a calculation. For example, consider the following calculation:
2 + 3 × 4 − 5
This problem could be solved simply by working from left to right—adding the 2 and 3, then multiplying by 4, and then subtracting 5:
2 + 3 × 4 − 5 =
5 × 4 − 5 =
20 − 5 = 15
It could be solved another way by multiplying the 3 and 4 and then adding the 2 and subtracting the 5:
2 + 3 × 4 − 5 =
2 + 12 − 5 =
14 − 5 = 9
The two methods give different answers—15 and 9. But for the sake of clarity, only one answer should be correct.
The order of operations is a set of rules that mathematicians use to avoid such confusing multiple answers. These rules tell which operations (kinds of calculations) to perform first, regardless of the order in which the numbers and signs are written. The first three rules are:
1. First, handle any powers and roots.
2. Next, do any multiplication and division.
3. Finally, perform any addition and subtraction.
According to these rules, the second method used in the calculation example—multiplying before adding—is correct. So, the correct answer is 9.
Within each “step” in the order of operations, it does not matter which particular operation is done first. In the previous calculation, for example, the 2 can be added before or after subtracting 5. As long as the multiplication is done first, the result is the same.
Powers and roots occur in more complex math problems, for example:
7 + 5 × 23 − √9 × 4
The raised 3 after the 2 means the third power of 2—in other words, two multiplied by itself three times or 2 × 2 × 2, which equals 8. This operation must be performed before multiplying by the 5. Similarly, the radical symbol (√) over the 9 means the square root of 9—that is, the number that results in 9 when multiplied by itself. That number is 3. This operation must be performed before multiplying by the 4. The correct answer to the above problem is 35.
Sometimes, the writer of a math problem wants to make an exception to the normal order of operation. The exception can be expressed using grouping symbols, such as parentheses or brackets. The use of grouping symbols adds an additional rule to the order of operations:
4. Whatever is inside grouping symbols, such as parenthesis or brackets, must be done first.
For an example, consider the original problem with the addition of some parentheses:
(2 + 3) × 4 − 5
The parentheses mean to add the 2 and 3 together first, even before any multiplication. Thus, the correct answer of the modified problem is 15.
Any fraction bar in a problem should be treated much as if parentheses surrounded everything above and below it. That is, the operations both above and below the bar should be done before dividing with the bar. For example, consider:
8 + 6 / 2 + 5
Before dividing, the 8 must be added to the 6. Likewise, the 2 must be added to the 5. Thus, the correct answer is 2.
The same principle also applies for radical signs that extend over multiple numbers. For example, consider:
√32 + 7 × 2
Everything inside the radical sign (32 + 7) must be done before taking the square root. The correct answer to the problem is 8.
