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Section 3.1 Simplifying Expressions and Combining Like Terms 133 2. Commutative, Associative, and Distributive Properties Several important properties of whole numbers were introduced in Sections 1.3 and 1.5 involving addition and multiplication of whole numbers. These properties also hold for integers and algebraic expressions and will be used to simplify expressions (Tables 3-1 through 3-3). Table 3-1 Commutative Properties Property In Symbols/Examples Comments/Notes Commutative property a b b a The order in which two of addition numbers are added does ex: 4 7 7 142 not affect the sum. x 3 3 x Commutative property a b b a The order in which two of multiplication numbers are multiplied ex: does not affect the product. 5 9 9 152 8y y 8 Table 3-2 Associative Properties Property In Symbols/Examples Comments/Notes Associative property 1a b2 c a 1b c2 The manner in which three of addition numbers are grouped under ex: 15 82 1 5 18 12 addition does not affect 1t n2 3 t 1n 32 the sum. Associative property 1a b2 c a 1b c2 The manner in which three of multiplication numbers are grouped under ex: 12 32 6 2 13 62 multiplication does not 3 1m n2 13 m2 n affect the product. Table 3-3 Distributive Property of Multiplication over Addition Property In Symbols/Examples Comments/Notes Distributive property a 1b c2 a b a c The factor outside the of multiplication over parentheses is multiplied by addition* ex: 312 x23122 1321x2 each term in the sum. 6 132x = 6 3x *Note that the distributive property of multiplication over addition is sometimes referred to as the distributive property. Avoiding Mistakes In an expression such as x 3, the two terms x and 3 cannot be combined into one term because they are not like terms. Addition of like terms will be discussed in Examples 6 and 7. c. The terms 6x and 4x are like terms. d. 3x2y and 5xy2 are not like terms because the exponents on the x variables are different, and the exponents on the y variables are different.


miller_prealgebra_2e_ch1_3
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