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Chapter 01 – Section 06 Commutative and Associative Properties
© William James Calhoun 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
© William James Calhoun 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. For any numbers a and b, a + b = b + a and a * b = b * a. COMMUTATIVE PROPERTY You can add/multiply two numbers in any order. 7 7 no Same as adding.
© William James Calhoun 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. For any numbers a, b, and c, (a + b) + c = a + (b + c) and ab(c) = a(bc). ASSOCIATIVE PROPERTY 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.
© William James Calhoun NOTE: Associative and Commutative Properties do NOT work for subtraction and division! Here is a chart of the properties mainly used to simplify expressions.
© William James Calhoun EXAMPLE 1α: Simplify each expression. a. 6(a + b) – a + 3bb. Distribute. Reorder (if you want.) Signs in front stay the same. CLT Clean up, if necessary. 6a + 6b – a + 3b 6a – a + 6b + 3b 5a + 9b
© William James Calhoun EXAMPLE 1β: Simplify each expression. a. 6(2x + 4y) + 2(x + 9)b. 5(0.3x + 0.1y) + 0.2x
© William James Calhoun 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. Remember: Your main goal should be to employ these properties. There is no need to memorize their proper names. 3 2 t++t 2 2+ 2 + t 2 + t 2 + 3 t 2 + t 2 + 2 + 3 2t 2 + 5
© William James Calhoun 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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