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Understanding Commutative and Associative Properties in Algebra

This section focuses on the commutative and associative properties, crucial for simplifying mathematical expressions. The commutative property states that for any numbers (a) and (b), (a + b = b + a) and (a cdot b = b cdot a); the order of addition or multiplication does not change the result. The associative property asserts that for any numbers (a), (b), and (c), ((a + b) + c = a + (b + c)) and (a(bc) = (ab)c); the way we group numbers does not affect their sum or product. Essential for problem-solving in algebra, these properties enhance understanding and efficiency in calculations.

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Understanding Commutative and Associative Properties in Algebra

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  1. Chapter 01 – Section 06 Commutative and Associative Properties

  2. 1-6 COMMUTATIVE AND ASSOCIATIVE PROPERTIES OBJECTIVES To recognize and use the commutative and associative properties and to simplify expressions. This section is about two more properties. You do need to memorize their names, but Focus on understanding how these properties work with problems. Remember that SIMPLIFY means: 1) Get rid of all parenthesis (distribution) 2) Combine Like Terms

  3. COMMUTATIVE PROPERTY For any numbers a and b, a + b = b + a and a * b = b * a. 1-6 COMMUTATIVE AND ASSOCIATIVE PROPERTIES • Question: What is 2 + 5? • How about 5 + 2? • Does it matter in what order you add two numbers? • How about three numbers? • Question: What about multiplying numbers? • commutative property- the order in which you add or multiply two numbers does not change their sum or product. 7 7 no no Same as adding. You can add/multiply two numbers in any order.

  4. ASSOCIATIVE PROPERTY For any numbers a, b, and c, (a + b) + c = a + (b + c) and ab(c) = a(bc). 1-6 COMMUTATIVE AND ASSOCIATIVE PROPERTIES • Question: Is there any mathematical difference to these two statements? • Two plus five is seven. Seven plus four is eleven. • Five plus four is nine. Nine plus two is eleven. • No – both give you eleven in the end. • associate property - the way you group three or more numbers when adding or multiplying does NOT change their sum or product. When adding and multiplying - group any way you want to. 2 + 3 + 5 = (2 + 3) + 5 = 2 + (3 + 5) = 10 and 2 * 3 * 5 = (2 * 3) * 5 = 2 * (3 * 5) = 30.

  5. 1-6 COMMUTATIVE AND ASSOCIATIVE PROPERTIES NOTE: Associative and Commutative Properties do NOT work for subtraction and division! Here is a chart of the properties mainly used to simplify expressions.

  6. 1-6 COMMUTATIVE AND ASSOCIATIVE PROPERTIES EX1β EXAMPLE 1α: Simplify each expression. a. 6(a + b) – a + 3b b. 6a + 6b – a + 3b 6a – a + 6b + 3b 5a + 9b 5a + 9b Distribute. Reorder (if you want.) Signs in front stay the same. CLT Clean up, if necessary.

  7. 1-6 COMMUTATIVE AND ASSOCIATIVE PROPERTIES EXAMPLE 1β: Simplify each expression. a. 6(2x + 4y) + 2(x + 9) b. 5(0.3x + 0.1y) + 0.2x

  8. 2 + t 2 + t 2 + 3 1-6 COMMUTATIVE AND ASSOCIATIVE PROPERTIES EX2β EXAMPLE 2α: a. Write an algebraic expression for the verbal expression the sum of two and the square of t increased by the sum of t squared and 3. b. Then simplify the algebraic expression. 2 + t2 + t2 + 3 t2 + t2 + 2 + 3 2t2 + 5 Remember: Your main goal should be to employ these properties. There is no need to memorize their proper names.

  9. 1-6 COMMUTATIVE AND ASSOCIATIVE PROPERTIES EXAMPLE 2β: a. Write an algebraic expression for the verbal expression the difference of x cubed and three increased by the difference of 5 and x cubed. b. Then simplify the algebraic expression.

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