 # Commutative and Associative Properties

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Commutative and Associative Properties
Chapter 1 Section 3 Commutative and Associative Properties

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

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

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)

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)

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

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

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

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

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.

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

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

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.

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

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.

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.

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