1.2.3 Simplifying Expressions

Key Objectives • Use the Commutative and Associative Properties. • Use the Distributive Property with mental math. • Combine like terms to simplify algebraic expressions • Use properties to justify simplification steps. Key Terms • The Commutative Property is the property that states that two or more values can be added or multiplied together in any order without changing the result. a + b = b + a a · b = b · a • The Associative Property is the property that states that values being added or multiplied together can be grouped in any order without changing the result. ( a + b ) + c = a + ( b + c ) ( a · b ) · c = a · ( b · c ) • The Distributive Property is the property that states that the product of a value and a sum (or difference) is equal to that value multiplied by each of the values being added (or subtracted). a ( b + c ) = ab + ac a ( b − c ) = ab − ac • A term can be a number, a variable, or a product of numbers and variables. Terms in an expression are separated by plus or minus signs. • Like terms are two or more terms that have the same variable raised to the same exponent. Example 1 Using the Commutative and Associative Properties By the Commutative Property, two or more numbers can be added in any order, and two or more numbers can be multiplied in any order, without changing the result. For example, 8 + 1 = 1 + 8 and 8 · 1 = 1 · 8. By the Associative Property, three or more numbers being added can be grouped in any way, and three or more numbers being multiplied can be grouped in any way, without changing the result. For example, 2 + (3 + 5) = (2 + 3) + 5 and 2 · (3 · 5) = (2 · 3) · 5. In Example 1, Prof. Burger demonstrates using the Commutative and Associative Properties to reorder and regroup parts of an expression, resulting in calculations that are easy to perform using mental math. COMMON ERROR ALERT The Commutative and Associative Properties cannot be applied to subtraction or division. For example, 8 ÷ (4 ÷ 2) = 8 ÷ 2 = 4, while (8 ÷ 4) ÷ 2 = 2 ÷ 2 = 1. Therefore, the Associative Property does not hold true for division. The same is true for subtraction.

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