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# Order of Operations

## The order of operations outlines the order by which certain operatives should take place in a mathematical expression for it to remain true

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PEMDAS

What if your teacher asked you to evaluate the expression \begin{align*}3+2 \times 6 \div (3-1)\end{align*}? Which should you do first, the addition, subtraction, multiplication, or division? What should you do second, third, and fourth? Also, should the parentheses affect your decisions? After completing this Concept, you'll be able to answer these questions and correctly evaluate the expression to your teacher's delight!

### Guidance

The Mystery of Math Verbs

Some math verbs are “stronger” than others and must be done first. This method is known as the order of operations.

A mnemonic (a saying that helps you remember something difficult) for the order of operations is PEMDAS- Please Excuse My Daring Aunt Sophie.

The order of operations:

Whatever is found inside PARENTHESES must be done first. EXPONENTS are to be simplified next. MULTIPLICATION and DIVISION are equally important and must be performed moving left to right. ADDITION and SUBTRACTION are also equally important and must be performed moving left to right.

#### Example A

Use the order of operations to simplify \begin{align*}(7-2) \times 4 \div 2-3\end{align*}.

Solution: First, we check for parentheses. Yes, there they are and must be done first.

Next we look for exponents (little numbers written a little above the others). No, there are no exponents so we skip to the next math verb.

Multiplication and division are equally important and must be done from left to right.

Finally, addition and subtraction are equally important and must be done from left to right.

\begin{align*}10-3 = 7\end{align*}

#### Example B

Use the order of operations to simplify the following expressions.

a) \begin{align*}3 \times 5-7 \div 2\end{align*}

b) \begin{align*}3 \times (5 - 7) \div 2\end{align*}

c) \begin{align*}(3 \times 5) - (7 \div 2)\end{align*}

Solutions:

a) There are no parentheses and no exponents. Go directly to multiplication and division from left to right: \begin{align*} 3 \times 5 - 7 \div 2 = 15 - 7 \div 2 = 15 - 3.5\end{align*}

Now subtract: \begin{align*}15 - 3.5 = 11.5\end{align*}

b) Parentheses must be done first: \begin{align*}3 \times (-2) \div 2\end{align*}

There are no exponents, so multiplication and division come next and are done left to right: \begin{align*}3 \times (-2) \div 2 = -6 \div 2 = -3\end{align*}

c) Parentheses must be done first: \begin{align*}(3 \times 5) - (7 \div 2) = 15 - 3.5\end{align*}

There are no exponents, multiplication, division, or addition, so simplify:

Parentheses are used two ways. The first is to alter the order of operations in a given expression, such as example (b). The second way is to clarify an expression, making it easier to understand.

Some expressions contain no parentheses, while others contain several sets of parentheses. Some expressions even have parentheses inside parentheses, called nested parentheses! Always start at the innermost parentheses and work outward.

#### Example C

Use the order of operations to evaluate the following expression when

.

\begin{align*}(2x-3)+x^2-3\end{align*}

Solution:

First, we will substitute in 2 for \begin{align*}x\end{align*}.

\begin{align*}(2(2)-3)+(2)^2-3\end{align*}

Now we will use the order of operations to evaluate the expression, starting inside the parentheses and then with the exponent.

\begin{align*}(2(2)-3)+(2)^2-3=(1)+4-3=2\end{align*}

We finish evaluating with addition and subtraction.

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### Guided Practice

1. Use the order of operations to simplify

\begin{align*}8-[19-(2+5)-7]\end{align*}.

2. Use the order or operations to evaluate the following expression when

and
.

\begin{align*}3\cdot y^2 -2(7-x)\end{align*}

Solutions:

1. Begin with the innermost parentheses:

Simplify according to the order of operations:

2. First, we will substitute in 3 for \begin{align*}x\end{align*} and 5 for \begin{align*}y\end{align*}.

Now we will use the order of operations to evaluate the expression, doing parentheses and exponents first, then multiplication, and finally subtraction.

Note that there was no division or addition, so we skipped those steps.

### Explore More

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Order of Operations (14:23)

Use the order of operations to simplify the following expressions.

1. \begin{align*}8 - (19 - (2 + 5) - 7)\end{align*}
2. \begin{align*}2 + 7 \times 11 - 12 \div 3\end{align*}
3. \begin{align*}(3 + 7) \div (7 - 12)\end{align*}
4. \begin{align*}8 \cdot 5 + 6^2\end{align*}
5. \begin{align*}9 \div 3 \times 7 - 2^3 + 7\end{align*}
6. \begin{align*}8 + 12 \div 6 + 6\end{align*}
7. \begin{align*}(7^2-3^2) \div 8\end{align*}

Evaluate the following expressions involving variables.

1. \begin{align*}2y^2\end{align*} when \begin{align*}y = 5\end{align*}
2. \begin{align*}3x^2 + 2x + 1\end{align*} when \begin{align*}x = 5\end{align*}
3. \begin{align*}(y^2 - x)^2\end{align*} when \begin{align*}x = 2\end{align*} and \begin{align*}y = 1\end{align*}

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 1.3.

### Vocabulary Language: English Spanish

nested parentheses

nested parentheses

Nested parentheses describe groups of terms inside of other groups. By convention, nested parentheses may be identified with other grouping symbols, such as the braces "{}" and brackets "[]" in the expression $\{ 3 + [ 2 - ( 5 + 4 ) ] \}$. Always evaluate parentheses from the innermost set outward.
order of operations

order of operations

A set of rules that tells you the order in which to perform operations.

Addition is an operation used to combine groups of like terms.
Brackets

Brackets

Brackets [ ], are symbols that are used to group numbers in mathematics. Brackets are the 'second level' of grouping symbols, used to enclose items already in parentheses.
division

division

Division is a simplified form of repeated subtraction. Division is used to determine the number of times that one term may be subtracted from another before reaching zero. Phrases such as 'the quotient of', 'divided equally', and 'per' all mean to use division or to divide.
Grouping Symbols

Grouping Symbols

Grouping symbols are parentheses or brackets used to group numbers and operations.
multiplication

multiplication

Multiplication is a simplified form of repeated addition. Multiplication is used to determine the result of adding a term to itself a specified number of times.
Parentheses

Parentheses

Parentheses "(" and ")" are used in algebraic expressions as grouping symbols.
PEMDAS

PEMDAS

PEMDAS (Please Excuse My Daring Aunt Sally) is a mnemonic device used to help remember the order of operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.
Real Number

Real Number

A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.
subtraction

subtraction

Subtraction is an operation used to determine the difference between values. It is the same as adding the opposite, the additive inverse, of a number. Words such as the difference between, minus, decrease, less, fewer, loss all mean to use subtraction.