People need a common set of rules for performing basic calculations. What does \(\ 3+5 \cdot 2\) equal? Is it 16 or 13? Your answer depends on how you understand the order of operations — a set of rules that tell you the order in which addition, subtraction, multiplication, and division are performed in any calculation.
Mathematicians have developed a standard order of operations that tells you which calculations to make first in an expression with more than one operation. Without a standard procedure for making calculations, two people could get two different answers to the same problem.
The building blocks of the order of operations are the arithmetic operations: addition, subtraction, multiplication, and division. The order of operations states:
What is the correct answer for the expression \(\ 3+5 \cdot 2\)? Use the order of operations listed above.
Multiply first. \(\ 3+5 \cdot 2=3+10\)
Then add. \(\ 3+10=13\)
This order of operations is true for all real numbers.
Simplify \(\ 7-5+3 \cdot 8\).
Solution
| \(\ 7-5+3 \cdot 8\) | According to the order of operations, multiplication comes before addition and subtraction. Multiply \(\ 3 \cdot 8\). |
| \(\ 7-5+24\) | Now, add and subtract from left to right. \(\ 7-5\) comes first. |
| \(\ 2+24=26\) | Finally, add \(\ 2+24\). |
When you are applying the order of operations to expressions that contain fractions, decimals, and negative numbers, you will need to recall how to do these computations as well.
Simplify: \(\ 3 \cdot \frac-8 \div \frac\)
Solution
| \(\ 3 \cdot \frac-8 \div \frac\) | According to the order of operations, multiplication comes before addition and subtraction. Multiply \(\ 3 \cdot \frac\) first. |
| \(\ 1-8 \div \frac\) | Now, divide \(\ 8 \div \frac\). |
| \(\ 8 \div \frac=\frac \cdot \frac=32\) | |
| \(\ 1-32=-31\) | Subtract. |
When you are evaluating expressions, you will sometimes see exponents used to represent repeated multiplication. Recall that an expression such as \(\ 7^\) is exponential notation for \(\ 7 \cdot 7\). (Exponential notation has two parts: the base and the exponent or the power. In \(\ 7^\), 7 is the base and 2 is the exponent; the exponent determines how many times the base is multiplied by itself.)
Exponents are a way to represent repeated multiplication; the order of operations places it before any other multiplication, division, subtraction, and addition is performed.
Simplify: \(\ 3^ \cdot 2^\)
Solution
| \(\ 3^ \cdot 2^\) | This problem has exponents and multiplication in it. According to the order of operations, simplifying \(\ 3^\) and \(\ 2^\) comes before multiplication. |
| \(\ 9 \cdot 2^\) | \(\ 3^\) is \(\ 3 \cdot 3\), which equals \(\ 9\). |
| \(\ 9 \cdot 8\) | \(\ 2^\) is \(\ 2 \cdot 2 \cdot 2\), which equals \(\ 8\). |
| \(\ 9 \cdot 8=72\) | Multiply. |
Simplify: \(\ \left(\frac\right)^+\left(\frac\right)^ \cdot 32\)
Solution
| \(\ \left(\frac\right)^+\left(\frac\right)^ \cdot 32\) | This problem has exponents, multiplication, and addition in it. According to the order of operations, simplify the terms with the exponents first, then multiply, then add. |
| \(\ \frac+\left(\frac\right)^ \cdot 32\) | Evaluate: \(\ \left(\frac\right)^=\frac \cdot \frac=\frac\) |
| \(\ \frac+\frac \cdot 32\) | Evaluate: \(\ \left(\frac\right)^=\frac \cdot \frac \cdot \frac=\frac\) |
| \(\ \frac+\frac\) | Multiply. |
| \(\ \frac+\frac=\frac\) | Simplify. \(\ \frac=\frac\), so you can add \(\ \frac+\frac\). |
Simplify: \(\ 100-5^ \cdot 4\)
The final piece that you need to consider in the order of operations is grouping symbols. These include parentheses \(\ (\quad)\), brackets \(\ [\quad]\), braces \(\ \<\quad\>\), and even fraction bars. These symbols are often used to help organize mathematical expressions (you will see them a lot in algebra).
Grouping symbols are used to clarify which operations to do first, especially if a specific order is desired. If there is an expression to be simplified within the grouping symbols, follow the order of operations.
When there are grouping symbols within grouping symbols, calculate from the inside to the outside. That is, begin simplifying within the innermost grouping symbols first.
Remember that parentheses can also be used to show multiplication. In the example that follows, both uses of parentheses—as a way to represent a group, as well as a way to express multiplication—are shown.
Simplify: \(\ (3+4)^+(8)(4)\)
Solution
| \(\ (3+4)^+(8)(4)\) | This problem has parentheses, exponents, multiplication, and addition in it. The first set of parentheses is a grouping symbol. The second set indicates multiplication. |
| \(\ (3+4)^+(8)(4)\) | Grouping symbols are handled first. Add numbers in parentheses. |
| \(\ 7^+(8)(4)\) | Simplify \(\ 7^\). |
| \(\ 49+\bf(8)(4)\) | Perform multiplication. |
| \(\ 49+32=81\) | Perform addition. |
Simplify: \(\ (1.5+3.5)-2(0.5 \cdot 6)^\)
Solution
Grouping symbols are handled first. Add numbers in the first set of parentheses.
Multiply numbers in the second set of parentheses.
Solution
This problem has brackets, parentheses, fractions, exponents, multiplication, subtraction, and addition in it.
Grouping symbols are handled first. The parentheses around the -6 aren’t a grouping symbol, they are simply making it clear that the negative sign belongs to the 6. Start with the innermost set of parentheses that are a grouping symbol, here it is in the numerator of the fraction, (\(\ 2 \cdot-6\)), and begin working out. (The fraction line acts as a type of grouping symbol, too; you simplify the numerator and denominator independently, and then divide the numerator by the denominator at the end.)
The order of operations is important to know, but it is sometimes hard to remember. Some people use a saying to help them remember the order of operations. This saying is “Please Excuse My Dear Aunt Sally,” or PEMDAS for short. The first letter of each word begins with the same letter of an arithmetic operation.
The P in Please stands for Parentheses (and other grouping symbols).
The E in Excuse stands for Exponents.
The M and D in My Dear stand for Multiplication and Division (from left to right).
The A and S in Aunt Sally stand for Addition and Subtraction (from left to right).
Note: Even though multiplication comes before division in the saying, division could be performed first. Whether multiplication or division is performed first is determined by which comes first when reading from left to right. The same is true of addition and subtraction. Don’t let the saying confuse you about this!
Absolute value expressions are one final method of grouping that you may see. Recall that the absolute value of a quantity is always positive or 0.
When you see an absolute value expression included within a larger expression, follow the regular order of operations and evaluate the expression within the absolute value sign. Then take the absolute value of that expression. The example below shows how this is done.
Solution
This problem has absolute values, decimals, multiplication, subtraction, and addition in it.
Grouping symbols, including absolute value, are handled first. Simplify the numerator, then the denominator. Evaluate \(\ |2-6|\).
