Section 1.6 Properties of Real Numbers and Simplifying Expressions 101 Notice that the same result is obtained if the factor of 6 is multiplied by each of the numbers 2 and 3, and then their products are added: The factor of 6 is distributed to the numbers 2 and 3. 612 32 6122 6132 12 18 30 The distributive property of multiplication over addition states that this is true in general. Distributive Property of Multiplication over Addition If a, b, and c are real numbers, then a1b c2 ab ac and 1b c2a ab ac Applying the Distributive Property Apply the distributive property: Solution: Apply the distributive property. Simplify. 21a 6b 72 21a 6b 72 21a2 216b2 2172 Because the difference addition as a 1b2, of two expressions can be written in terms of the distributive property can be applied when the operation of subtraction is present within the parentheses. For example: 51y 72 53y 1724 7 Rewrite subtraction as addition of . Apply the distributive property. 5 3y 1724 51y2 5172 5y 1352, or 5y 35 Simplify. a b 2a 12b 14 21a 6b 72 Example 5 TIP: The mathematical definition of the distributive property is consistent with the everyday meaning of the word distribute. To distribute means to “spread out from one to many.” In the mathematical context, the factor a is distributed to both b and c in the parentheses. Skill Practice 14. Apply the distributive property. 71x 4y z2 Classroom Example: p. 107, Exercise 44 TIP: Notice that the parentheses are removed after the distributive property is applied. Sometimes this is referred to as clearing parentheses. Answer 14. 7x 28y 7z
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