262 CHAPTER 5 Number Theory and the Real Number System Earlier we stated that variables are letters used to represent numbers. In algebra, we will often use the properties of real numbers to simplify expressions involving variables. When a number is placed directly to the left of a variable, it indicates multiplication. For example, x5 means ⋅ x 5 . We will use variables in the next example. Example 2 Identifying Properties of Real Numbers Name the property illustrated. a) + = + 5 2 2 5 b) + + = + + x x ( 9) 4 (9 4) c) = ⋅ y y 8(7 ) (8 7) d) + = ⋅ ⋅ + ⋅ s s 3( 2) 3 3 2 e) + + = + + x x 6 (4 ) 6 ( 4) f) ⋅ = ⋅ r r (4 ) 2 2 (4 ) Solution a) Commutative property of addition b) Associative property of addition c) Associate property of multiplication d) Distributive property of multiplication over addition e) The only change between the left and right sides of the equal sign is the order of the 4 and x within the parentheses. The order is changed from + x 4 to + x 4 using the commutative property of addition. f) The order of the r (4 ) and 2 is changed using the commutative property of multiplication. 7 Now try Exercise 41 Example 3 Simplifying by Using the Distributive Property Simplify. a) + 4(5 6) b) + 2(3 5) Solution a) 4(5 6) (4 5) (4 6) 20 4 6 + = ⋅ + ⋅ = + b) + = ⋅ + ⋅ = + 2(3 5) (2 3) (2 5) 3 2 10 Note that ⋅ 2 3 is written as 3 2. 7 Now try Exercise 61 Example 4 Distributive Property Use the distributive property to multiply + x 2( 5). Then simplify the result. Solution x x x 2( 5) 2 2 5 2 10 + = ⋅ + ⋅ = + 7 Now try Exercise 53 Instructor Resources for Section 5.5 in MyLab Math • Objective-Level Videos 5.5 • PowerPoint Lecture Slides 5.5 • MyLab Exercises and Assignments 5.5 We summarize the properties mentioned in this section as follows, where a, b, and c are any real numbers.
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