Order of Operations
1.1 Learning Objectives
- Recognize and combine like terms in an expression
- Use the order of operations to simplify expressions
- Simplify expressions with fraction bars, brackets, and parentheses
- Use the distributive property to simplify expressions with grouping symbols
- Simplify expressions containing absolute values
Why use the order of operations?
What is [latex]3+5\times2[/latex]? 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! Some important terminology before we begin:- operations/operators: In mathematics we call things like multiplication, division, addition, and subtraction operations. They are the verbs of the math world, doing work on numbers and variables. The symbols used to denote operations are called operators, such as [latex]+{, }-{, }\times{, }\div[/latex]. As you learn more math, you will learn more operators.
- term: Examples of terms would be [latex]2x[/latex] and [latex]-\frac{3}{2}[/latex] or [latex]a^3[/latex]. Even lone integers can be a term, like 0.
- expression: A mathematical expression is one that connects terms with mathematical operators. For example [latex]\frac{1}{2}+\left(2^2\right)- 9\div\frac{6}{7}[/latex] is an expression.
1.1.1 Combining Like Terms
One way we can simplify expressions is to combine like terms. Like terms are terms where the variables match exactly (exponents included). Examples of like terms would be [latex]5xy[/latex] and [latex]-3xy[/latex] or [latex]8a^2b[/latex] and [latex]a^2b[/latex] or [latex]-3[/latex] and [latex]8[/latex]. If we have like terms we are allowed to add (or subtract) the numbers in front of the variables, then keep the variables the same. As we combine like terms we need to interpret subtraction signs as part of the following term. This means if we see a subtraction sign, we treat the following term like a negative term. The sign always stays with the term. This is shown in the following examples:Example 1.1.A
Combine like terms: [latex]5x-2y-8x+7y[/latex]Answer: The like terms in this expression are:
[latex]5x[/latex] and [latex]-8x[/latex]
[latex]-2y[/latex] and [latex]7y[/latex]
Note how we kept the sign in front of each term.
Combine like terms:
[latex]5x-8x = -3x[/latex]
[latex]-2y+7y = 5y[/latex]
Note how signs become operations when you combine like terms.
Simplified Expression:
[latex]5x-2y-8x+7y=-3x+5y[/latex]
Example 1.1.B
Combine like terms: [latex]x^2-3x+9-5x^2+3x-1[/latex]Answer: The like terms in this expression are:
[latex]x^2[/latex] and [latex]-5x^2[/latex]
[latex]-3x[/latex] and [latex]3x[/latex]
[latex]9[/latex] and [latex]-1[/latex]
Combine like terms:
[latex]\begin{array}{r}x^2-5x^2 = -4x^2\\-3x+3x=0\,\,\,\,\,\,\,\,\,\,\,\\9-1=8\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]
Simplified Expression:
[latex]x^2-3x+9-5x^2+3x-1=-4x^2+8[/latex]
1.1.2 Order of Operations
You may or may not recall the order of operations for applying several mathematical operations to one expression. Just as it is a social convention for us to drive on the right-hand side of the road, the order of operations is a set of conventions used to provide order when you are required to use several mathematical operations for one expression.The Order of Operations
- Perform all operations within grouping symbols first. Grouping symbols include parentheses ( ), brackets [ ], braces { }, and fraction bars.
- Evaluate exponents or square roots.
- Multiply or divide, from left to right.
- Add or subtract, from left to right.
Example 1.1.3
Simplify [latex]7–5+3\cdot8[/latex].Answer: According to the order of operations, multiplication comes before addition and subtraction. Multiply [latex]3\cdot8[/latex].
[latex]\begin{array}{c}7–5+3\cdot8\\7–5+24\end{array}[/latex]
Now, add and subtract from left to right. [latex]7–5[/latex] comes first.[latex]2+24[/latex].
Finally, add.
[latex]2+24=26[/latex]
Answer
[latex-display]7–5+3\cdot8=26[/latex-display]Example 1.1.C
Simplify [latex]3\cdot\frac{1}{3}-8\div\frac{1}{4}[/latex].Answer: According to the order of operations, multiplication and division come before addition and subtraction. Sometimes it helps to add parentheses to help you know what comes first, so let's put parentheses around the multiplication and division since it will come before the subtraction.
[latex]\left(3\cdot\frac{1}{3}\right)-\left(8\div\frac{1}{4}\right)[/latex]
Multiply [latex] 3\cdot \frac{1}{3}[/latex] first.[latex]\left( 3\cdot \frac{1}{3}\right)-\left(8\div \frac{1}{4}\right)[/latex]
[latex]\left(1\right)-\left(8\div \frac{1}{4}\right)[/latex]
Now, divide [latex]8\div\frac{1}{4}[/latex].
[latex]8\div\frac{1}{4}=\frac{8}{1}\cdot\frac{4}{1}=32[/latex]
Subtract.[latex]\left(1\right)–\left(32\right)=−31[/latex]
Answer
[latex-display] 3\cdot \frac{1}{3}-8\div \frac{1}{4}=-31[/latex-display]1.1.3 Exponents and Square Roots
In this section, we expand our skills with applying the order of operation rules to expressions with exponents and square roots. If the expression has exponents or square roots, they are to be performed after parentheses and other grouping symbols have been simplified and before any multiplication, division, subtraction, and addition that are outside the parentheses or other grouping symbols. Recall that an expression such as [latex]7^{2}[/latex] is exponential notation for [latex]7\cdot7[/latex]. (Exponential notation has two parts: the base and the exponent or the power. In [latex]7^{2}[/latex], 7 is the base and 2 is the exponent; the exponent determines how many times the base is multiplied by itself.) Exponents are a way to represent repeated multiplication; the order of operations places it before any other multiplication, division, subtraction, and addition is performed.Example 1.1.D
Simplify [latex]3^{2}\cdot2^{3}[/latex].Answer: This problem has exponents and multiplication in it. According to the order of operations, simplifying [latex]3^{2}[/latex] and [latex]2^{3}[/latex] comes before multiplication.
[latex]3^{2}\cdot2^{3}[/latex]
[latex] {{3}^{2}}[/latex] is [latex]3\cdot3[/latex], which equals 9.[latex] 9\cdot {{2}^{3}}[/latex]
[latex] {{2}^{3}}[/latex] is [latex]2\cdot2\cdot2[/latex], which equals 8.[latex] 9\cdot 8[/latex]
Multiply.[latex] 9\cdot 8=72[/latex]
Answer
[latex-display] {{3}^{2}}\cdot {{2}^{3}}=72[/latex-display]Example 1.1.E
Simplify [latex]\left(3+4\right)^{2}+\left(8\right)\left(4\right)[/latex].Answer: This problem has parentheses, exponents, multiplication, and addition in it. The first set of parentheses is a grouping symbol. The second set indicates multiplication. Grouping symbols are handled first. Add numbers in parentheses.
[latex]\begin{array}{c}(3+4)^{2}+(8)(4)\\(7)^{2}+(8)(4)\end{array}[/latex]
Simplify [latex]7^{2}[/latex].[latex]\begin{array}{c}7^{2}+(8)(4)\\49+(8)(4)\end{array}[/latex]
Multiply.
[latex]\begin{array}{c}49+(8)(4)\\49+(32)\end{array}[/latex]
Add.
[latex]49+32=81[/latex]
Answer
[latex-display](3+4)^{2}+(8)(4)=81[/latex-display]Example 1.1.F
Simplify [latex]\Large\frac{\sqrt{7+2}+2^2}{(8)(4)-11}[/latex].Answer: This problem has all the operations to consider with the order of operations. Grouping symbols are handled first, in this case the fraction bar. We will simplify the top and bottom separately. To simplify the top:
[latex]\sqrt{7+2}+2^2[/latex]
Add the numbers inside the square root, simplify the result and [latex]2^2[/latex][latex]\begin{array}{c}\sqrt{7+2}+2^2\\\\=\sqrt{9}+4\\\\=3+4=7\end{array}[/latex]
To simplify the bottom:[latex](8)(4)-11[/latex]
Multiply 8 and 4 first, then subtract 11.[latex](8)(4)-11=32-11=21[/latex]
Now put the fraction back together to see if any more simplifying needs to be done. [latex-display]\Large\frac{7}{21}[/latex], this can be reduced to [latex]\Large\frac{1}{3}[/latex-display]Answer
[latex]\Large\frac{\sqrt{7+2}+2^2}{(8)(4)-11}=\frac{1}{3}[/latex]
Think About It
These problems are very similar to the examples given above. How are they different and what tools do you need to simplify them? a) Simplify [latex]\left(1.5+3.5\right)–2\left(0.5\cdot6\right)^{2}[/latex]. This problem has parentheses, exponents, multiplication, subtraction, and addition in it, as well as decimals instead of integers. Use the box below to write down a few thoughts about how you would simplify this expression with decimals and grouping symbols. [practice-area rows="2"][/practice-area]
Answer: Grouping symbols are handled first. Add numbers in the first set of parentheses.
[latex]\begin{array}{c}(1.5+3.5)–2(0.5\cdot6)^{2}\\5–2(0.5\cdot6)^{2}\end{array}[/latex]
Multiply numbers in the second set of parentheses.[latex]\begin{array}{c}5–2(0.5\cdot6)^{2}\\5–2(3)^{2}\end{array}[/latex]
Evaluate exponents.[latex]\begin{array}{c}5–2(3)^{2}\\5–2\cdot9\end{array}[/latex]
Multiply.[latex]\begin{array}{c}5–2\cdot9\\5–18\end{array}[/latex]
Subtract.[latex]5–18=−13[/latex]
Answer
[latex-display](1.5+3.5)–2(0.5\cdot6)^{2}=−13[/latex-display]b) Simplify [latex] {{\left( \frac{1}{2} \right)}^{2}}+{{\left( \frac{1}{4} \right)}^{3}}\cdot \,32[/latex].
Use the box below to write down a few thoughts about how you would simplify this expression with fractions and grouping symbols. [practice-area rows="2"][/practice-area]Answer: This problem has exponents, multiplication, and addition in it, as well as fractions instead of integers. According to the order of operations, simplify the terms with the exponents first, then multiply, then add.
[latex]\left(\frac{1}{2}\right)^{2}+\left(\frac{1}{4}\right)^{3}\cdot32[/latex]
Evaluate: [latex]\left(\frac{1}{2}\right)^{2}=\frac{1}{2}\cdot\frac{1}{2}=\frac{1}{4}[/latex][latex]\frac{1}{4}+\left(\frac{1}{4}\right)^{3}\cdot32[/latex]
Evaluate: [latex]\left(\frac{1}{4}\right)^{3}=\frac{1}{4}\cdot\frac{1}{4}\cdot\frac{1}{4}=\frac{1}{64}[/latex][latex]\frac{1}{4}+\frac{1}{64}\cdot32[/latex]
Multiply.[latex] \frac{1}{4}+\frac{32}{64}[/latex]
Simplify. [latex] \frac{32}{64}=\frac{1}{2}[/latex], so you can add [latex] \frac{1}{4}+\frac{1}{2}[/latex].[latex] \frac{1}{4}+\frac{1}{2}=\frac{3}{4}[/latex]
Answer
[latex] {{\left( \frac{1}{2} \right)}^{2}}+{{\left( \frac{1}{4} \right)}^{3}}\cdot 32=\frac{3}{4}[/latex]1.1.4 Multiple Operations
Next, we will simplify mathematical expressions that contain many grouping symbols and many operations. We will also use the distributive property to break up multiplication into addition. Additionally, you will see how to handle absolute value terms when you simplify expressions.Example 1.1.G
Simplify [latex]\Large\frac{5-[3+(2\cdot (-6))]}{{{3}^{2}}+2}[/latex]Answer: This problem has brackets, parentheses, fractions, exponents, multiplication, subtraction, and addition in it. Grouping symbols are handled first. The parentheses around the [latex]-6[/latex] aren’t a grouping symbol; they are simply making it clear that the negative sign belongs to the 6. Start with the innermost set of parentheses that are a grouping symbol. In this example, the innermost set of parentheses would be in the numerator of the fraction, [latex](2\cdot(−6))[/latex]. Begin working out from there. (The fraction line acts as a type of grouping symbol, too; you simplify the numerator and denominator independently, and then divide the numerator by the denominator at the end.)
[latex]\Large\begin{array}{c}\frac{5-\left[3+\left(2\cdot\left(-6\right)\right)\right]}{3^{2}+2}\\\\\frac{5-\left[3+\left(-12\right)\right]}{3^{2}+2}\end{array}[/latex]
Add [latex]3[/latex] and [latex]-12[/latex], which are in brackets, to get [latex]-9[/latex].[latex]\Large\begin{array}{c}\frac{5-\left[3+\left(-12\right)\right]}{3^{2}+2}\\\\\frac{5-\left[-9\right]}{3^{2}+2}\end{array}[/latex]
Subtract [latex]5–\left[−9\right]=5+9=14[/latex].[latex]\Large\begin{array}{c}\frac{5-\left[-9\right]}{3^{2}+2}\\\\\frac{14}{3^{2}+2}\end{array}[/latex]
The top of the fraction is all set, but the bottom (denominator) has remained untouched. Apply the order of operations to that as well. Begin by evaluating [latex]3^{2}=9[/latex].[latex]\Large\begin{array}{c}\frac{14}{3^{2}+2}\\\\\frac{14}{9+2}\end{array}[/latex]
Now add. [latex]9+2=11[/latex].[latex]\Large\begin{array}{c}\frac{14}{9+2}\\\\\frac{14}{11}\end{array}[/latex]
Answer
[latex-display]\Large\frac{5-\left[3+\left(2\cdot\left(-6\right)\right)\right]}{3^{2}+2}=\frac{14}{11}[/latex-display]1.1.5 The Distributive Property
Parentheses are used to group, or combine expressions and terms in mathematics. You may see them used when you are working with formulas, and when you are translating a real situation into a mathematical problem so you can find a quantitative solution.
Combo Meal Distributive Property[latex]\begin{array}{c}\,\,\,3\left(2\text{ tacos }+ 1 \text{ drink}\right)\\=3\cdot{2}\text{ tacos }+3\text{ drinks }\\\,\,=6\text{ tacos }+3\text{ drinks }\end{array}[/latex]
The distributive property allows us to explicitly describe a total that is a result of a group of groups. In the case of the combo meals, we have three groups of ( two tacos plus one drink). The following definition describes how to use the distributive property in general terms.
The Distributive Property of Multiplication
For all real numbers a, b, and c, [latex]a(b+c)=ab+ac[/latex]. What this means is that when a number multiplies an expression inside parentheses, you can distribute the multiplication to each term of the expression individually.multiply three by (the sum of three and y), then subtract y, then add 9
To multiply three by the sum of three and y, you use the distributive property -
[latex]\begin{array}{c}\,\,\,\,\,\,\,\,\,3\left(3+y\right)-y+9\\\,\,\,\,\,\,\,\,\,=\underbrace{3\cdot{3}}+\underbrace{3\cdot{y}}-y+9\\=9+3y-y+9\end{array}[/latex]
Now you can subtract y from 3y and add 9 to 9.
[latex]\begin{array}{c}9+3y-y+9\\=18+2y\end{array}[/latex]
The next example shows how to use the distributive property when one of the terms involved is negative.
Example 1.1.H
Simplify [latex]a+2\left(5-b\right)+3\left(a+4\right)[/latex]Answer: This expression has two sets of parentheses with variables locked up in them. We will use the distributive property to remove the parentheses.
[latex]\begin{array}{c}a+2\left(5-b\right)+3\left(a+4\right)\\=a+2\cdot{5}-2\cdot{b}+3\cdot{a}+3\cdot{4}\end{array}[/latex]
Note how we placed the negative sign that was on b in front of the 2 when we applied the distributive property. When you multiply a negative by a positive the result is negative, so [latex]2\cdot{-b}=-2b[/latex]. It is important to be careful with negative signs when you are using the distributive property.
[latex]\begin{array}{c}a+2\cdot{5}-2\cdot{b}+3\cdot{a}+3\cdot{4}\\=a+10-2b+3a+12\\=4a+22-2b\end{array}[/latex]
We combined all the terms we could to get our final result.
Answer
[latex-display]a+2\left(5-b\right)+3\left(a+4\right)=4a+22-2b[/latex-display]Absolute Value
Absolute value expressions are one final method of grouping that you may see. Recall that the absolute value of a quantity is always positive or 0. When you see an absolute value expression included within a larger expression, treat the absolute value like a grouping symbol and evaluate the expression within the absolute value sign first. Then take the absolute value of that expression. The example below shows how this is done.Example 1.1.I
Simplify [latex]\Large\frac{3+\left|2-6\right|}{2\left|3\cdot1.5\right|-\left(-3\right)}[/latex].Answer: This problem has absolute values, decimals, multiplication, subtraction, and addition in it. Grouping symbols, including absolute value, are handled first. Simplify the numerator, then the denominator. Evaluate [latex]\left|2–6\right|[/latex].
[latex]\Large\begin{array}{c}\frac{3+\left|2-6\right|}{2\left|3\cdot1.5\right|-\left(-3\right)}\\\\\frac{3+\left|-4\right|}{2\left|3\cdot1.5\right|-\left(-3\right)}\end{array}[/latex]
Take the absolute value of [latex]\left|−4\right|[/latex].[latex]\Large\begin{array}{c}\frac{3+\left|-4\right|}{2\left|3\cdot1.5\right|-\left(-3\right)}\\\\\frac{3+4}{2\left|3\cdot1.5\right|-\left(-3\right)}\end{array}[/latex]
Add the numbers in the numerator.[latex]\Large\begin{array}{c}\frac{3+4}{2\left|3\cdot1.5\right|-\left(-3\right)}\\\\\frac{7}{2\left| 3\cdot 1.5 \right|-(-3)}\end{array}[/latex]
Now that the numerator is simplified, turn to the denominator. Evaluate the absolute value expression first. [latex]3 \cdot 1.5 = 4.5[/latex], giving[latex]\Large\begin{array}{c}\frac{7}{2\left|{3\cdot{1.5}}\right|-(-3)}\\\\\frac{7}{2\left|{ 4.5}\right|-(-3)}\end{array}[/latex]
The expression “[latex]2\left|4.5\right|[/latex]” reads “2 times the absolute value of 4.5.” Multiply 2 times 4.5.[latex]\Large\begin{array}{c}\frac{7}{2\left|4.5\right|-\left(-3\right)}\\\\\frac{7}{9-\left(-3\right)}\end{array}[/latex]
Subtract.[latex]\Large\begin{array}{c}\frac{7}{9-\left(-3\right)}\\\\\frac{7}{12}\end{array}[/latex]
Answer
[latex-display]\Large\frac{3+\left|2-6\right|}{2\left|3\cdot1.5\right|-3\left(-3\right)}=\frac{7}{12}[/latex-display]Summary
The order of operations gives us a consistent sequence to use in computation. Without the order of operations, you could come up with different answers to the same computation problem. (Some of the early calculators, and some inexpensive ones, do NOT use the order of operations. In order to use these calculators, the user has to input the numbers in the correct order.)Licenses & Attributions
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- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
- ScreenHot: Combo Meal Image. Provided by: Lumen Learning License: CC BY: Attribution.
- Ex 2: Combining Like Terms. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
- Order of Operations with a Fraction Containing a Square Root. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
CC licensed content, Shared previously
- Simplify an Expression in Fraction Form. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Simplify an Expression in Fraction Form with Absolute Values. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Unit 9: Real Numbers, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education Located at: https://www.nroc.org/. License: CC BY: Attribution.
- Ex 3: Combining Like Terms Requiring Distribution. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Simplify an Expression in the Form: a-b+c*d. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Simplify an Expression in the Form: a*1/b-c/(1/d)`. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Simplify an Expression in the Form: a^n*b^m. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Simplify an Expression in the Form: (a+b)^2+c*d. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
