Evaluating and Simplifying Expressions Using the Commutative and Associative Properties
Learning Outcomes
- Evaluate algebraic expressions for a given value using the commutative and associative properties of addition and multiplication
- Simplify algebraic expressions using the commutative and associative properties of addition and multiplication
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].
- [latex]x+0.37+\left(-x\right)[/latex]
- [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
[ohm_question]145792[/ohm_question]
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]
Answer:
Solution:
| 1. |
|
|
[latex]\Large\frac{4}{3}(\frac{3}{4}\normalsize n\Large)[/latex] |
| Substitute 17 for n. |
[latex]\Large\frac{4}{3}(\frac{3}{4}\normalsize\cdot\color{red}{17}\Large)[/latex] |
| Multiply in the parentheses first. |
[latex]\Large\frac{4}{3}(\frac{51}{4})[/latex] |
| Multiply again. |
[latex]17[/latex] |
| 2. |
|
|
[latex]\Large(\frac{4}{3}\normalsize\cdot\Large\frac{3}{4})\normalsize n[/latex] |
| Substitute 17 for n. |
[latex]\Large(\frac{4}{3}\normalsize\cdot\Large\frac{3}{4})\normalsize\cdot\color{red}{17}[/latex] |
| Multiply. The product of reciprocals is 1. |
[latex](1)\cdot17[/latex] |
| Multiply again. |
[latex]17[/latex] |
What was the difference between part 1 and part 2 here? Only the grouping changed. By the Associative Property of Multiplication, [latex]\Large\frac{4}{3}\left(\Large\frac{3}{4}\normalsize n\Large\right)\normalsize =\Large\left(\frac{4}{3}\normalsize\cdot\Large\frac{3}{4}\right)\normalsize n[/latex]. By carefully choosing how to group the factors, we can make the work easier.
try it
[ohm_question]145796[/ohm_question]
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]
Answer:
Solution:
Notice the first and third terms are opposites, so we can use the commutative property of addition to reorder the terms.
|
[latex]-84n+\left(-73n\right)+84n[/latex] |
| Re-order the terms. |
[latex]-84n+84n+\left(-73n\right)[/latex] |
| Add left to right. |
[latex]0+\left(-73n\right)[/latex] |
| Add. |
[latex]-73n[/latex] |
try it
[ohm_question]145797[/ohm_question]
Watch the following video for more similar examples of how to use the associative and commutative properties to simplify expressions.
https://youtu.be/8vMRywgaqOE
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]
Answer:
Solution:
Notice the first and third terms are reciprocals, so we can use the Commutative Property of Multiplication to reorder the factors.
|
[latex]\Large\frac{7}{15}\cdot \frac{8}{23}\cdot \frac{15}{7}[/latex] |
| Re-order the terms. |
[latex]\Large\frac{7}{15}\cdot \frac{15}{7}\cdot \frac{8}{23}[/latex] |
| Multiply left to right. |
[latex]1\cdot\Large\frac{8}{23}[/latex] |
| Multiply. |
[latex]\Large\frac{8}{23}[/latex] |
try it
[ohm_question]145798[/ohm_question]
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]
Answer:
Solution:
Notice that the second and third terms have a common denominator, so this work will be easier if we change the grouping.
|
[latex]\Large\left(\frac{5}{13}\normalsize +\Large\frac{3}{4}\right)\normalsize +\Large\frac{1}{4}[/latex] |
| Group the terms with a common denominator. |
[latex]\Large\frac{5}{13}\normalsize +\Large\left(\frac{3}{4}\normalsize +\Large\frac{1}{4}\right)[/latex] |
| Add in the parentheses first. |
[latex]\Large\frac{5}{13}\normalsize +\Large\left(\frac{4}{4}\right)[/latex] |
| Simplify the fraction. |
[latex]\Large\frac{5}{13}\normalsize +1[/latex] |
| Add. |
[latex]1\Large\frac{5}{13}[/latex] |
| Convert to an improper fraction. |
[latex]\Large\frac{18}{13}[/latex] |
try it
[ohm_question]145799[/ohm_question]
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]
Answer:
Solution:
Notice that the sum of the second and third coefficients is a whole number.
|
[latex]\left(6.47q+9.99q\right)+1.01q[/latex] |
| Change the grouping. |
[latex]6.47q+\left(9.99q+1.01q\right)[/latex] |
| Add in the parentheses first. |
[latex]6.47q+\left(11.00q\right)[/latex] |
| Add. |
[latex]17.47q[/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]
try it
[ohm_question]145800[/ohm_question]
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]
Answer:
Solution:
|
[latex]6\left(9x\right)[/latex] |
| Use the associative property of multiplication to re-group. |
[latex]\left(6\cdot 9\right)x[/latex] |
| Multiply in the parentheses. |
[latex]54x[/latex] |
try it
[ohm_question]145973[/ohm_question]
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]
Answer:
Solution:
Use the Commutative Property of Addition to re-order so that like terms are together.
|
[latex]18p+6q+\left(-15p\right)+5q[/latex] |
| Re-order terms. |
[latex]18p+\left(-15p\right)+6q+5q[/latex] |
| Combine like terms. |
[latex]3p+11q[/latex] |
try it
Simplify: [latex]23r+14s+9r+\left(-15s\right)[/latex]
Answer:
[latex-display]32r−s[/latex-display]
Simplify: [latex]37m+21n+4m+\left(-15n\right)[/latex]
Answer:
[latex-display]41m+6n[/latex-display]
Licenses & Attributions
CC licensed content, Original
- Review Properties of Real Numbers While Simplifying Expressions. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Question ID 145792, 145796, 145797, 145798, 145799, 145973. Authored by: Lumen Learning. License: CC BY: Attribution.
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