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Study Guides > Intermediate Algebra

Simplify Expressions With the Order of Operations

Learning Outcomes

  • Recognize and combine like terms in an expression
  • Use the order of operations to simplify expressions

Introduction

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], [latex]-\dfrac{3}{2}[/latex], or [latex]a^3[/latex]. Even lone integers can be a term, like [latex]0[/latex].
  • expression: A mathematical expression is one that connects terms with mathematical operators. For example  [latex]\dfrac{1}{2}\normalsize +\left(2^2\right)- 9\div\dfrac{6}{7}[/latex] is an expression.

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. Kind of like saying four apples plus three apples equals seven apples. But two apples plus six oranges can't be combined and simplified because they are not "like terms". 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

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]

In the following video you will be shown how to combine like terms using the idea of the distributive property.  Note that this is a different method than is shown in the written examples on this page, but it obtains the same result. https://youtu.be/JIleqbO8Tf0

Example

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]-4x^2+8[/latex]

In the video that follows, you will be shown another example of combining like terms.  Pay attention to why you are not able to combine all three terms in the example. https://youtu.be/b9-7eu29pNM

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 a formulaic outline to follow 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.
This order of operations is true for all real numbers.

Example

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]

In the following example, you will be shown how to simplify an expression that contains both multiplication and subtraction using the order of operations. https://youtu.be/yFO_0dlfy-w When you are applying the order of operations to expressions that contain fractions, decimals, and negative numbers, you will need to recall how to do these computations as well.

Example

Simplify [latex]3\cdot\dfrac{1}{3}\normalsize -8\div\dfrac{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\dfrac{1}{3}\normalsize\right)-\left(8\div\dfrac{1}{4}\normalsize\right)[/latex]

Multiply [latex] 3\cdot\dfrac{1}{3}[/latex] first.

[latex]\left( 3\cdot\dfrac{1}{3}\normalsize\right)-\left(8\div\dfrac{1}{4}\normalsize\right)[/latex]

[latex]\left(1\right)-\left(8\div\dfrac{1}{4}\normalsize\right)[/latex]

Now, divide [latex]8\div\dfrac{1}{4}[/latex].

[latex]8\div\dfrac{1}{4}\normalsize =\dfrac{8}{1}\normalsize\cdot\dfrac{4}{1}\normalsize =32[/latex]

Subtract.

[latex]\left(1\right)–\left(32\right)=−31[/latex]

Answer

[latex-display] 3\cdot\dfrac{1}{3}\normalsize -8\div\dfrac{1}{4}\normalsize =-31[/latex-display]

In the following video you are shown how to use the order of operations to simplify an expression that contains multiplication, division, and subtraction with terms that contain fractions. https://youtu.be/yqp06obmcVc

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], [latex]7[/latex] is the base and [latex]2[/latex] 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

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 [latex]9[/latex].

[latex] 9\cdot {{2}^{3}}[/latex]

[latex] {{2}^{3}}[/latex] is [latex]2\cdot2\cdot2[/latex], which equals [latex]8[/latex].

[latex] 9\cdot 8[/latex]

Multiply.

[latex] 9\cdot 8=72[/latex]

Answer

[latex-display] {{3}^{2}}\cdot {{2}^{3}}=72[/latex-display]

In the video that follows, an expression with exponents on its terms is simplified using the order of operations. https://youtu.be/JjBBgV7G_Qw When there are grouping symbols within grouping symbols, calculate from the inside to the outside. That is, begin simplifying within the innermost grouping symbols first. Remember that parentheses can also be used to show multiplication. In the example that follows, both uses of parentheses—as a way to represent a group, as well as a way to express multiplication—are shown.

Example

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]

In the following video, you are shown how to use the order of operations to simplify an expression with grouping symbols, exponents, multiplication, and addition. https://youtu.be/EMch2MKCVdA In the next example we will simplify an expression that has a square root.

Example

Simplify [latex]\dfrac{\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 numerator (top) and denominator (bottom) separately. To simplify the top:

[latex]\sqrt{7+2}+2^2[/latex]

Add the numbers inside the square root (as they are essentially grouped by that symbol/operator), and the term [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 [latex]8[/latex] and [latex]4[/latex] first, then subtract [latex]11[/latex].

[latex](8)(4)-11=[/latex]

[latex]\hspace{1cm}32-11=21[/latex]

Now put the fraction back together to see if any more simplifying needs to be done. The simplified numerator equaled [latex]7[/latex], and the simplified denominator equaled [latex]21[/latex].

so [latex]\dfrac{7}{21}[/latex] , which can be reduced to [latex]\dfrac{1}{3}[/latex]

Answer

[latex]\dfrac{\sqrt{7+2}+2^2}{(8)(4)-11}=\dfrac{1}{3}[/latex]

https://youtu.be/9suc63qB96o

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(\dfrac{1}{2}\normalsize\right)}^{2}}+{{\left(\dfrac{1}{4}\normalsize\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(\dfrac{1}{2}\normalsize\right)^{2}+\left(\dfrac{1}{4}\normalsize\right)^{3}\cdot32[/latex]

Evaluate: [latex]\left(\dfrac{1}{2}\normalsize\right)^{2}=\dfrac{1}{2}\normalsize\cdot\dfrac{1}{2}\normalsize =\dfrac{1}{4}[/latex]

[latex]\dfrac{1}{4}\normalsize +\left(\dfrac{1}{4}\normalsize\right)^{3}\cdot32[/latex]

Evaluate: [latex]\left(\dfrac{1}{4}\normalsize\right)^{3}=\frac{1}{4}\normalsize\cdot\dfrac{1}{4}\normalsize\cdot\dfrac{1}{4}\normalsize=\dfrac{1}{64}[/latex]

[latex]\dfrac{1}{4}\normalsize +\dfrac{1}{64}\normalsize\cdot32[/latex]

Multiply.

[latex]\dfrac{1}{4}\normalsize +\dfrac{32}{64}[/latex]

Simplify. [latex]\dfrac{32}{64}\normalsize =\dfrac{1}{2}[/latex], so you can add [latex]\dfrac{1}{4}\normalsize +\dfrac{1}{2}[/latex].

[latex]\dfrac{1}{4}\normalsize +\dfrac{1}{2}\normalsize =\dfrac{3}{4}[/latex]

Answer

[latex] {{\left(\dfrac{1}{2}\normalsize\right)}^{2}}+{{\left(\dfrac{1}{4} \normalsize\right)}^{3}}\cdot 32=\dfrac{3}{4}[/latex]

Some people use a saying to help them remember the order of operations. This saying is called PEMDAS or Please Excuse My Dear Aunt Sally. The first letter of each word begins with the same letter of an arithmetic operation. Please [latex] \displaystyle \Rightarrow [/latex] Parentheses (and other grouping symbols) Excuse [latex] \displaystyle \Rightarrow [/latex] Exponents My Dear [latex] \displaystyle \Rightarrow [/latex] Multiplication and Division (from left to right) Aunt Sally [latex] \displaystyle \Rightarrow [/latex] Addition and Subtraction (from left to right) Note: Even though multiplication comes before division in the saying, division could be performed first. Which is performed first, between multiplication and division, is determined by which comes first when reading from left to right. The same is true of addition and subtraction. Don't let the saying confuse your understanding of this!

 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

CC licensed content, Original

  • 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

  • Ex 2: Combining Like Terms. 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.
  • 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.