Evaluating and Simplifying Expressions Using the Commutative and Associative Properties

Learning Outcomes

  • Evaluate expressions using the commutative and associative properties
  • Simplify expressions using the commutative and associative properties

Evaluate Expressions using the Commutative and Associative Properties

The commutative and associative properties can make it easier to evaluate some algebraic expressions. Since order does not matter when adding or multiplying three or more terms, we can rearrange and re-group terms to make our work easier, as the next several examples illustrate.

example

Evaluate each expression when [latex]x=\Large\frac{7}{8}[/latex].

  1. [latex]x+0.37+\left(-x\right)[/latex]
  2. [latex]x+\left(-x\right)+0.37[/latex]

Solution:

1.
[latex]x+0.37+(--x)[/latex]
Substitute [latex]\Large\frac{7}{8}[/latex] for [latex]x[/latex] . [latex]\color{red}{\Large\frac{7}{8}}\normalsize +0.37+(--\color{red}{\Large\frac{7}{8}})[/latex]
Convert fractions to decimals. [latex]0.875+0.37+(--0.875)[/latex]
Add left to right. [latex]1.245--0.875[/latex]
Subtract. [latex]0.37[/latex]
2.
[latex]x+(--x)+0.37[/latex]
Substitute [latex]\Large\frac{7}{8}[/latex] for x. [latex]\color{red}{\Large\frac{7}{8}}\normalsize +(--\color{red}{\Large\frac{7}{8}})+0.37[/latex]
Add opposites first. [latex]0.37[/latex]

What was the difference between part 1 and part 2? Only the order changed. By the Commutative Property of Addition, [latex]x+0.37+\left(-x\right)=x+\left(-x\right)+0.37[/latex]. But wasn’t part 2 much easier?

 

try it

Let’s do one more, this time with multiplication.

example

Evaluate each expression when [latex]n=17[/latex].
1. [latex]\Large\frac{4}{3}\left(\Large\frac{3}{4}\normalsize n\Large\right)[/latex]
2. [latex]\left(\Large\frac{4}{3}\normalsize\cdot\Large\frac{3}{4}\right)\normalsize n[/latex]

 

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Simplify Expressions Using the Commutative and Associative Properties

When we have to simplify algebraic expressions, we can often make the work easier by applying the Commutative or Associative Property first instead of automatically following the order of operations. Notice that in the first example, part 2 was easier to simplify than part 1 because the opposites were next to each other and their sum is [latex]0[/latex]. Likewise, part 2 in the second example was easier, with the reciprocals grouped together, because their product is [latex]1[/latex]. In the next few examples, we’ll use our number sense to look for ways to apply these properties to make our work easier.

example

Simplify: [latex]-84n+\left(-73n\right)+84n[/latex]

 

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Watch the following video for more similar examples of how to use the associative and commutative properties to simplify expressions.

Now we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals—their product is [latex]1[/latex].

example

Simplify: [latex]\Large\frac{7}{15}\cdot \frac{8}{23}\cdot \frac{15}{7}[/latex]

 

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In expressions where we need to add or subtract three or more fractions, combine those with a common denominator first.

example

Simplify: [latex]\Large\left(\frac{5}{13}\normalsize +\Large\frac{3}{4}\right)\normalsize +\Large\frac{1}{4}[/latex]

 

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When adding and subtracting three or more terms involving decimals, look for terms that combine to give whole numbers.

example

Simplify: [latex]\left(6.47q+9.99q\right)+1.01q[/latex]

Many people have good number sense when they deal with money. Think about adding [latex]99[/latex] cents and [latex]1[/latex] cent. Do you see how this applies to adding [latex]9.99+1.01?[/latex]

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When simplifying expressions that contain variables, we can use the commutative and associative properties to re-order or regroup terms, as shown in the next pair of examples.

example

Simplify: [latex]6\left(9x\right)[/latex]

 

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In The Language of Algebra, we learned to combine like terms by rearranging an expression so the like terms were together. We simplified the expression [latex]3x+7+4x+5[/latex] by rewriting it as [latex]3x+4x+7+5[/latex] and then simplified it to [latex]7x+12[/latex]. We were using the Commutative Property of Addition.

example

Simplify: [latex]18p+6q+\left(-15p\right)+5q[/latex]

 

try it

Simplify: [latex]23r+14s+9r+\left(-15s\right)[/latex]

Simplify: [latex]37m+21n+4m+\left(-15n\right)[/latex]

The following video provides more examples of how to simplify expressions using the commutative and associative properties of multiplication and addition.