[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.
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.
4.
Addition and
Subtraction
- 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.
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 parentheses? 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] |
example
Simplify: [latex]18\div 6+4\left(5 - 2\right)[/latex].
Answer:
Solution:
|
[latex]18\div 6+4(5-2)[/latex] |
| Parentheses? Yes, subtract first. |
[latex]18\div 6+4(\color{red}{3})[/latex] |
| Exponents? No. |
|
| Multiplication or division? Yes. |
|
| Divide first because we multiply and divide left to right. |
[latex]\color{red}{3}+4(3)[/latex] |
| Any other multiplication or division? Yes. |
|
| Multiply. |
[latex]3+\color{red}{12}[/latex] |
| Any other multiplication or division? No. |
|
| Any addition or subtraction? Yes. |
[latex]15[/latex] |
example
[latex]\text{Simplify: }5+{2}^{3}+3\left[6 - 3\left(4 - 2\right)\right][/latex].
Answer:
Solution:
|
[latex]5+{2}^{3}+ 3[6-3(4-2)][/latex] |
| Are there any parentheses (or other grouping symbol)? Yes. |
|
| Focus on the parentheses that are inside the brackets. |
[latex]5+{2}^{3}+ 3[6-3\color{red}{(4-2)}][/latex] |
| Subtract. |
[latex]5+{2}^{3}+3[6-\color{red}{3(2)}][/latex] |
| Continue inside the brackets and multiply. |
[latex]5+{2}^{3}+3[6-\color{red}{6}][/latex] |
| Continue inside the brackets and subtract. |
[latex]5+{2}^{3}+3[\color{red}{0}][/latex] |
| The expression inside the brackets requires no further simplification. |
|
| Are there any exponents? Yes. |
|
| Simplify exponents. |
[latex]5+\color{red}{{2}^{3}}+3[0][/latex] |
| Is there any multiplication or division? Yes. |
|
| Multiply. |
[latex]5+8+\color{red}{3[0]}[/latex] |
| Is there any addition or subtraction? Yes. |
|
| Add. |
[latex]\color{red}{5+8}+0[/latex] |
| Add. |
[latex]\color{red}{13+0}[/latex] |
|
[latex]13[/latex] |
example
Simplify: [latex]{2}^{3}+{3}^{4}\div 3-{5}^{2}[/latex].
Answer:
Solution:
|
[latex]{2}^{3}+{3}^{4}\div 3-{5}^{2}[/latex] |
| If an expression has several exponents, they may be simplified in the same step. |
|
| Simplify exponents. |
[latex]\color{red}{{2}^{3}}+\color{red}{{3}^{4}}\div 3-\color{red}{{5}^{2}}[/latex] |
| Divide. |
[latex]8+\color{red}{82\div 3}-25[/latex] |
| Add. |
[latex]\color{red}{8+27}-25[/latex] |
| Subtract. |
[latex]\color{red}{35-25}[/latex] |
|
[latex]10[/latex] |
