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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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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? 


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.

Use the order of operations to simplify the following expressions:

  1.  \begin{align*}(7-2) \times 4 \div 2-3.\end{align*}

First, check for parentheses. Since there are parentheses, they must be done first.

\begin{align*}(7 - 2) \times 4 \div 2 - 3 = (5) \times 4 \div 2 - 3\end{align*}

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

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

\begin{align*}5 \times 4 \div 2 - 3 & = 20 \div 2 - 3\\ 20 \div 2 - 3 & = 10 - 3\end{align*}

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

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

The answer is 7.

  1. \begin{align*}3 \times 5-7 \div 2\end{align*}

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*}

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

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*}

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

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:

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


Parentheses are used two ways. The first is to alter the order of operations in a given expression, such as in problem 3 and 4 from above. 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.

Use the order of operations to evaluate the following expression:
\begin{align*}(2 \cdot 2-3)+2^2-3\end{align*}

Use the order of operations to evaluate the expression, starting inside the parentheses, then evaluating the exponent, and finally adding and subtracting left to right.

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


Example 1

Earlier, you were asked to evaluate the expression \begin{align*}3+2 \times 6 \div (3-1)\end{align*}

You cannot just evaluate this expression in any order that you want. Instead, to evaluate this expression, you must use the order of operations (PEMDAS).

The first step is to simplify the expression inside the parentheses:

\begin{align*}3+2\times6\div (3-1)=3+2\times6\div 2\end{align*}

There are no exponents so the next step is multiplication and division from left to right:

\begin{align*}3+2\times6\div 2=3+12\div 2 \end{align*} \begin{align*}3+12\div 2=3+6\end{align*} Finally, add the two numbers together:

\begin{align*}3+6=9\end{align*} The expression is equal to 9. 

Example 2

Use the order of operations to simplify \begin{align*}8-[19-(2+5)-7].\end{align*} 

Begin with the innermost parentheses:


Simplify according to the order of operations:


Example 3

Use the order or operations to evaluate the following expression:

\begin{align*}3\cdot 5^2 -2(7-3)\end{align*}

Use the order of operations to evaluate the expression, doing parentheses and exponents first, then multiplication, and finally subtraction.

\begin{align*}3\cdot 5^2 -2(7-3) & = 3\cdot 5^2 - 2(4)\\ & = 3\cdot 25-2(4)\\ & = 75-8\\ &=67\end{align*}

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


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*}
  8. \begin{align*}2\cdot 5^2\end{align*}
  9. \begin{align*}3(5)^2 + 2(5) + 1\end{align*}
  10. \begin{align*}(1^2 - 2)^2\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 1.3. 

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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

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 [ ], 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 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 are parentheses or brackets used to group numbers and operations.


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


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

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


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.

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