Learning Outcomes
- Use the order of operations to simplify arithmetic expressions
Key words
- Arithmetic operation: [latex]+, -, \times, \div, a^{b}, \sqrt{}[/latex], etc.
- Arithmetic term: a number, or numbers that are multiplied, divided, or raised to a power
- Arithmetic expression: a term, or terms separated by addition and/or subtraction
- Order of operations: the order in which arithmetic operations are calculated
- Grouping symbol: symbols used to group terms [latex](\,), [\, ],\left\{\right\},\sqrt{\, }[/latex], etc.
Correctly using the order of operations
What is [latex]3+5\times2[/latex] ? Is it [latex]13[/latex] or [latex]16[/latex] ? 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 us 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, [latex]13[/latex] or [latex]16[/latex] ? By the end of this module you’ll know!
We need to clarify the order in which operations like [latex]+, -, \times, \div, a^{b}, \sqrt{}[/latex], etc. will be carried out. Otherwise, expressions may have different meanings, and they may result in different values.
For example, consider the expression:
[latex]4+3\cdot 7[/latex]
[latex]\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}[/latex]
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. Grouping Symbols: parentheses, brackets, braces, etc.
- Simplify all expressions inside parentheses or other grouping symbols, working on the innermost parentheses first.
2. Exponents
- Simplify all expressions with exponents.
3. Multiplication and Division from left to right
- Perform all multiplication and division in order from left to right. These operations have equal priority.
4. Addition and Subtraction
- Perform all addition and subtraction in order from left to right. These operations have equal priority.
example
Simplify the expressions:
- [latex]4+3\cdot 7[/latex]
- [latex]\left(4+3\right)\cdot 7[/latex]
Solution:
| 1. | |
| [latex]4+3\cdot 7[/latex] | |
| Are there any grouping symbols? No. | |
| Are there any exponents? No. | |
| Is there any multiplication or division? Yes. | |
| Multiply first. | [latex]4+\color{red}{3\cdot 7}[/latex] |
| Add. | [latex]4+21[/latex] |
| [latex]25[/latex] |
| 2. | |
| [latex](4+3)\cdot 7[/latex] | |
| Are there any parentheses? Yes. | [latex]\color{red}{(4+3)}\cdot 7[/latex] |
| Simplify inside the parentheses. | [latex](7)7[/latex] |
| Are there any exponents? No. | |
| Is there any multiplication or division? Yes. | |
| Multiply. | [latex]49[/latex] |
try it
example
Simplify:
- [latex]\text{18}\div \text{9}\cdot \text{2}[/latex]
- [latex]\text{18}\cdot \text{9}\div \text{2}[/latex]
try it
Another way to simplify expressions is to simply each term in an expression to a single number, then add the terms. A term is a number, or numbers that are multiplied, divided, raised to a power, etc., but not added or subtracted.
For example, the expression [latex]5^{2}+7\cdot (-3)+(6\cdot4)\div12[/latex] is an expression with three terms separated by [latex]+[/latex] signs:
[latex]5^{2}[/latex] simplifies to [latex]25[/latex]
[latex]7\cdot (-3)[/latex] simplifies to [latex]-21[/latex]
[latex](6\cdot4)\div12[/latex] simplifies to [latex]24\div12=2[/latex].
Then we add the terms: [latex]25+(-21)+2=6[/latex].
If there is a subtraction in the expression, it is best to write the expression as addition of the opposite. That way the subtraction sign gets attached to the term immediately following it. For example. [latex](-4)^{2}-5\cdot 4=(-4)^{2}+(-5\cdot 4)=16+(-20)=-4[/latex].
example
Simplify: [latex]18\div 6-4\left(5 - 2\right)[/latex].
example
Simplify: [latex]6\sqrt{25}+3^{2}\cdot (-3)-2^{3}\cdot 4[/latex]
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
[latex]\text{Simplify: }5+{2}^{3}+3\left[6 - 3\left(4 - 2\right)\right][/latex].
Solution
There are three terms. Use the order of operations on each term.
Color is used to show exactly what part of the expression is being evaluated at each step.
[latex]\begin{aligned}&5+\color{blue}{{2}^{3}}+3\left[6 - 3\color{green}{\left(4 - 2\right)}\right]\\&=5+\color{blue}{8}+3\left[6 - 3\color{green}{\left(2\right)}\right]\\&=5+8+3\left[6 - \color{green}{3\left(2\right)}\right]\\&=5+8+3\left[6 - \color{green}{6}\right]\\&=5+8+3\left[6 - \color{green}{6}\right]\\&=5+8+3\color{green}{\left[6 - 6\right]}\\&=5+8+\color{green}{3\left[0\right]}\\&=5+8+\color{green}{0}\\&=13\end{aligned}[/latex]
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: [latex]{2}^{3}+{3}^{4}\div 3-{5}^{2}[/latex].
