## Simplifying Expressions Using the Order of Operations

### Learning Outcomes

• Use the order of operations to simplify simple mathematical expressions
• Simplify complex mathematical expressions involving addition, subtraction, multiplication, division, and exponents

## Correctly using the order of operations

What is $3+5\times2$ ? Is it $13$ or $16$ ? This may seem like a trick question, but there is actually only one correct answer.

Many years ago, mathematicians developed a standard order of operations that tells you which calculations to make first in an expression with more than one operation. In other words, order of operations simply refers to the specific order of steps you should follow when you solve a math expression. Without a standard procedure for making calculations, two people could get two different answers to the same problem, like the one above. So which is it, $13$ or $16$ ? By the end of this module you’ll know!

We’ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order in which they will be carried out. Otherwise, expressions may have different meanings, and they may result in different values.

For example, consider the expression:

$4+3\cdot 7$

$\begin{array}{cccc}\hfill \text{Some students say it simplifies to 49.}\hfill & & & \hfill \text{Some students say it simplifies to 25.}\hfill \\ \begin{array}{ccc}& & \hfill 4+3\cdot 7\hfill \\ \text{Since }4+3\text{ gives 7.}\hfill & & \hfill 7\cdot 7\hfill \\ \text{And }7\cdot 7\text{ is 49.}\hfill & & \hfill 49\hfill \end{array}& & & \begin{array}{ccc}& & \hfill 4+3\cdot 7\hfill \\ \text{Since }3\cdot 7\text{ is 21.}\hfill & & \hfill 4+21\hfill \\ \text{And }21+4\text{ makes 25.}\hfill & & \hfill 25\hfill \end{array}\hfill \end{array}$

Imagine the confusion that could result if every problem had several different correct answers. The same expression should give the same result. So mathematicians established some guidelines called the order of operations, which outlines the order in which parts of an expression must be simplified.

### Order of Operations

When simplifying mathematical expressions perform the operations in the following order:
1. Parentheses and other Grouping Symbols

• Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.

2. Exponents

• Simplify all expressions with exponents.

3. Multiplication and Division

• Perform all multiplication and division in order from left to right. These operations have equal priority.

• Perform all addition and subtraction in order from left to right. These operations have equal priority.

Students often ask, “How will I remember the order?” Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase. Please Excuse My Dear Aunt Sally or PEMDAS.

Order of Operations
Excuse Exponents
My Dear Multiplication and Division

It’s good that ‘My Dear’ goes together, as this reminds us that multiplication and division have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.
Similarly, ‘Aunt Sally’ goes together and so reminds us that addition and subtraction also have equal priority and we do them in order from left to right.

### example

Simplify the expressions:

1. $4+3\cdot 7$
2. $\left(4+3\right)\cdot 7$

Solution:

 1. $4+3\cdot 7$ Are there any parentheses? No. Are there any exponents? No. Is there any multiplication or division? Yes. Multiply first. $4+\color{red}{3\cdot 7}$ Add. $4+21$ $25$
 2. $(4+3)\cdot 7$ Are there any parentheses? Yes. $\color{red}{(4+3)}\cdot 7$ Simplify inside the parentheses. $(7)7$ Are there any exponents? No. Is there any multiplication or division? Yes. Multiply. $49$

### example

Simplify:

1. $\text{18}\div \text{9}\cdot \text{2}$
2. $\text{18}\cdot \text{9}\div \text{2}$

### example

Simplify: $18\div 6+4\left(5 - 2\right)$.

### try it

In the video below we show another example of how to use the order of operations to simplify a mathematical expression.

When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.

### example

$\text{Simplify: }5+{2}^{3}+3\left[6 - 3\left(4 - 2\right)\right]$.

### try it

In the video below we show another example of how to use the order of operations to simplify an expression that contains exponents and grouping symbols.

### example

Simplify: ${2}^{3}+{3}^{4}\div 3-{5}^{2}$.

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