Mathematics » Properties of Real Numbers » Commutative and Associative Properties

Evaluating Expressions using the Commutative and Associative Properties

Evaluating 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 \(x=\frac{7}{8}.\)

  1. \(\phantom{\rule{0.2em}{0ex}}x+0.37+\left(-x\right)\)
  2. \(\phantom{\rule{0.2em}{0ex}}x+\left(-x\right)+0.37\)

Solution

(1) 
 Evaluating Expressions using the Commutative and Associative Properties
Substitute \(\frac{7}{8}\) for \(x\).Evaluating Expressions using the Commutative and Associative Properties
Convert fractions to decimals.Evaluating Expressions using the Commutative and Associative Properties
Add left to right.Evaluating Expressions using the Commutative and Associative Properties
Subtract.Evaluating Expressions using the Commutative and Associative Properties
(2) 
 Evaluating Expressions using the Commutative and Associative Properties
Substitute \(\frac{7}{8}\) for x.Evaluating Expressions using the Commutative and Associative Properties
Add opposites first.Evaluating Expressions using the Commutative and Associative Properties

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?

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

Example

Evaluate each expression when \(n=17.\)

  1. \(\phantom{\rule{0.2em}{0ex}}\frac{4}{3}\left(\frac{3}{4}n\right)\)
  2. \(\phantom{\rule{0.2em}{0ex}}\left(\frac{4}{3}·\frac{3}{4}\right)n\)

Solution

(1) 
 Evaluating Expressions using the Commutative and Associative Properties
Substitute 17 for n.Evaluating Expressions using the Commutative and Associative Properties
Multiply in the parentheses first.Evaluating Expressions using the Commutative and Associative Properties
Multiply again.Evaluating Expressions using the Commutative and Associative Properties
(2) 
 Evaluating Expressions using the Commutative and Associative Properties
Substitute 17 for n.Evaluating Expressions using the Commutative and Associative Properties
Multiply. The product of reciprocals is 1.Evaluating Expressions using the Commutative and Associative Properties
Multiply again.Evaluating Expressions using the Commutative and Associative Properties

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.

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