<h2>Evaluating Expressions using the Commutative and Associative Properties</h2>
<p>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.</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<p>Evaluate each expression when \(x=\frac{7}{8}.\)</p>
<ol type="1">
<li>\(\phantom{\rule{0.2em}{0ex}}x+0.37+\left(-x\right)\)</li>
<li>\(\phantom{\rule{0.2em}{0ex}}x+\left(-x\right)+0.37\)</li>
</ol>
</div>
<div>
<h3>Solution</h3>
<table data-label="" style="width: 601px;" summary="The image shows the expression x plus 0.37 plus negative x. Substitute the fraction 7 eights for x and the expression becomes 7 eights plus 0.37 plus negative 7 eighths. Convert the fractions to decimals to get 0.875 plus 0.37 plus negative 0.875. Adding from left to right 0.875 plus 0.37 becomes 1.245 and the expression becomes 1.245 plus negative 0.875. Adding a negative is the same as subtraction so now the expression can be written as 1.245 minus 0.875. Perform the subtraction to get 0.37.">
<tbody>
<tr>
<td style="width: 368px;">(1)</td>
<td style="width: 219px;"> </td>
</tr>
<tr>
<td style="width: 368px;"> </td>
<td style="width: 219px;"><img alt="." src="https://data.ulearngo.com/assets/98feae59-e941-4c0b-baea-a98c8ebacf77"/></td>
</tr>
<tr>
<td style="width: 368px;">Substitute \(\frac{7}{8}\) for \(x\).</td>
<td style="width: 219px;"><img alt="." src="https://data.ulearngo.com/assets/a3223858-7a83-46b6-a45e-4a2b18d1214e"/></td>
</tr>
<tr>
<td style="width: 368px;">Convert fractions to decimals.</td>
<td style="width: 219px;"><img alt="." src="https://data.ulearngo.com/assets/167ff01a-8282-474d-aa6d-568e880793f7"/></td>
</tr>
<tr>
<td style="width: 368px;">Add left to right.</td>
<td style="width: 219px;"><img alt="." src="https://data.ulearngo.com/assets/d265a6cc-deec-436a-9beb-faa46a601f5e"/></td>
</tr>
<tr>
<td style="width: 368px;">Subtract.</td>
<td style="width: 219px;"><img alt="." src="https://data.ulearngo.com/assets/a7412c5c-36f1-45a4-926c-68917d7ef08f"/></td>
</tr>
</tbody>
</table>
<table data-label="" style="width: 466px;" summary="The image shows the expression x plus negative x plus 0.37. Substitute the fraction 7 eights for x and the expression becomes 7 eights plus negative 7 eighths plus 0.37. Add the opposites first so that only 0.37 is left.">
<tbody>
<tr>
<td style="width: 276px;">(2)</td>
<td style="width: 176px;"> </td>
</tr>
<tr>
<td style="width: 276px;"> </td>
<td style="width: 176px;"><img alt="." src="https://data.ulearngo.com/assets/32ba0daa-4bd4-4ab4-9197-9e0de50aa2cc"/></td>
</tr>
<tr>
<td style="width: 276px;">Substitute \(\frac{7}{8}\) for x.</td>
<td style="width: 176px;"><img alt="." src="https://data.ulearngo.com/assets/853fed0b-5721-4a3e-ab25-27512876cb30"/></td>
</tr>
<tr>
<td style="width: 276px;">Add opposites first.</td>
<td style="width: 176px;"><img alt="." src="https://data.ulearngo.com/assets/dd3fd16c-1177-47c1-8b3f-11ca4cab1b1f"/></td>
</tr>
</tbody>
</table>
<p>What was the difference between part and part ? Only the order changed. By the Commutative Property of Addition, \(x+0.37+\left(-x\right)=x+\left(-x\right)+0.37.\) But wasn’t part much easier?</p>
</div>
</div>
</div>
</div>
<p>Let’s do one more, this time with multiplication.</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<p>Evaluate each expression when \(n=17.\)</p>
<ol>
<li>\(\phantom{\rule{0.2em}{0ex}}\frac{4}{3}\left(\frac{3}{4}n\right)\)</li>
<li>\(\phantom{\rule{0.2em}{0ex}}\left(\frac{4}{3}·\frac{3}{4}\right)n\)</li>
</ol>
</div>
<div>
<h3>Solution</h3>
<table data-label="" summary="The image shows the expression 4 thirds times the quantity 3 quarters n in parentheses. Substitute 17 for n and the expression becomes 4 thirds times the quantity 3 quarters times 17 in parentheses. Simplify inside the parentheses first. Multiply 3 quarters by 17 to get 51 quarters. The expression becomes 4 thirds times the quantity 51 quarters in parentheses. Multiply to get 17.">
<tbody>
<tr>
<td>(1)</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td><img alt="." src="https://data.ulearngo.com/assets/73fb2cff-7299-4dfe-a6f9-feeed95eaa6e"/></td>
</tr>
<tr>
<td>Substitute 17 for n.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/279d4efb-e4d3-4879-8108-c59c6c9603f2"/></td>
</tr>
<tr>
<td>Multiply in the parentheses first.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/1fe7b26e-db4e-49ee-acf3-880211bfbe6e"/></td>
</tr>
<tr>
<td>Multiply again.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/d125b621-7ff6-425e-8791-98ec794195bd"/></td>
</tr>
</tbody>
</table>
<table data-label="" summary="The image shows the expression the quantity 4 thirds times 3 quarters in parentheses times n. Substitute 17 for n. Multiply inside the parentheses first to get 1. The equation is 1 times 17 to get 17.">
<tbody>
<tr>
<td>(2)</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td><img alt="." src="https://data.ulearngo.com/assets/ab4e3aae-9c41-4ad9-91e0-980c31b3b2f0"/></td>
</tr>
<tr>
<td>Substitute 17 for n.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/c462cf46-285f-4cdf-8633-380f2f45e291"/></td>
</tr>
<tr>
<td>Multiply. The product of reciprocals is 1.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/1c862141-7fcf-4c07-9da5-3ae19dae18da"/></td>
</tr>
<tr>
<td>Multiply again.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/eacdc266-0e45-4330-b454-c85fd599440a"/></td>
</tr>
</tbody>
</table>
<p>What was the difference between part and part here? Only the grouping changed. By the Associative Property of Multiplication, \(\frac{4}{3}\left(\frac{3}{4}n\right)=\left(\frac{4}{3}·\frac{3}{4}\right)n.\) By carefully choosing how to group the factors, we can make the work easier.</p>
</div>
</div>
</div>
</div>

77e14597-4d58-4399-be3f-b1bb5c28bc54

Evaluating Expressions using the Commutative and Associative Properties

Mathematics studies measurement, relationships, and properties of quantities and sets, using numbers and symbols. Arithmetic, algebra, geometry, and calculus are popular topics in mathematics.

Mathematics

Learn about the properties of real numbers including rational and irrational numbers, classifying real numbers, using the commutative, associative, distributive, identity, inverse, and zero properties in simplifying and evaluating expressions, and converting between different units of measurement and temperature scales.


Evaluating Expressions using the Commutative and Associative Properties

Track Your Learning Progress