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**Commutative and Associative Properties**

Chapter 1 Section 3 Commutative and Associative Properties

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**Commutative Property of Addition**

Words: The order in which two numbers are added does not change their sum. Numbers: = 7 + 5

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**Commutative Property of Multiplication**

Words: The order in which two numbers are multiplied does not change their product. Numbers: 5 ∙ 7 = 7 ∙ 5

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**Associative Property of Addition**

Words: The way in which three numbers are grouped when they are added does not change their sum. Numbers: (24 + 8) + 2 = 24 + (8 + 2)

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**Associative Property of Multiplication**

Words: The way in which three numbers are grouped when they are multiplied does not change their product. Numbers: (24 ∙ 8) ∙ 2 = 24 ∙ (8 ∙ 2)

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Example 1 Name the property shown by each statement. 4 ∙ 11 ∙ 2 = 11 ∙ 4 ∙ 2 Commutative Property of Multiplication

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Example 2 Name the property shown by each statement. (n + 12) + 5 = n + (12 + 5) Associative Property of Addition

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Your Turn Name the property shown for each statement. (5 ∙ 4) ∙ 3 = 5 ∙ (4 ∙ 3) Associative Property of Multiplication

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Your Turn Name the property shown for each statement t + 1 = t Commutative Property of Addition

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You can use these properties to simplify and evaluate algebraic expressions. To simplify an expression, eliminate all parentheses first and then add, subtract, multiply or divide.

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Example 3 Simplify the expression 15 + (3x + 8). Identify the properties used in each step. 15 + (3x + 8) = (15 + (8 + 3x) (15 + 8) + 3x 23 + 3x Commutative Property of Addition Associative Property of Addition Substitution Property

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**Commutative Property of Addition**

Your Turn Simplify each expression. Identify the properties used in each step a a a Commutative Property of Addition

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Example 4 The volume of a box can be found using the expression l ∙ w ∙ h, where l is the length, w is the width, and h is the height. Find the volume if a box whose length is 30 inches, width is 6 inches, and height is 5 inches. l ∙ w ∙ h = 30 ∙ 6 ∙ 5 = 30 ∙ (6 ∙ 5) Associative Property of (x) = 30 ∙ 30 Substitution Property = 90 Substitution Property The volume of the box is 900 cubic inches.

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**Associative Property of Multiplication**

Your Turn Simplify each expression. Identify the properties used in each step. (x ∙ 5) ∙ 20 x ∙ (5 ∙ 20) x ∙ x Associative Property of Multiplication

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**Closure Property of Whole Numbers**

Words: Because the sum of product of two whole numbers is also a whole number, the set of whole numbers is closed under addition and multiplication. Numbers: 2 + 5 = 7 and 7 is a whole number. 2 ∙ 5 = 10 and 10 is a whole number.

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Example 5 State whether the statement Division of whole numbers is commutative is true or false. If false, provide a counterexample. 6 ÷ 3 = 3 ÷ 6 Evaluate each expression separately. 2 ≠ ½ 6 ÷ 3 = 2 and 3 ÷ 6 = ½ We found a counterexample, so the statement is false. Division of whole numbers is not commutative.

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Your Turn State whether the statement Division of whole numbers is commutative is true or false. If false, provide a counterexample. (12 – 6) – 2 = 12 – (6 – 2) 6 – 2 = 12 – 4 4 = 8

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