. Operations and Properties. OBJECTIVES. Represent the four arithmetic operations using variables. Evaluate expressions using the order of operations. Recognize and apply the properties of addition 4. Recognize and apply the properties of multiplication 5. Recognize and apply the distributive property NOTE Francois Viete (540 60), a French mathematician, first introduced the practice of using letters to represent known and unknown quantities. NOTE We do not use a multiplication sign because of the possible confusion with the letter x. NOTE The symbols and first appeared in print in a book by Johann Widman (489). The symbol dates to a text by William Oughtred (6). The process of combining two elements of a set to produce a third element is called a binary operation. There are four basic binary operations: addition, subtraction, multiplication, and division. In algebra, we write these operations as follows: x y is called the sum of x and y, or x plus y. x y is called the difference of x and y, or x minus y. xy (or x y) is the product of x and y, or x times y. x y (or x y) is the quotient of x and y, or x divided by y. Each of the above is an example of an expression. An expression is a meaningful collection of numbers, variables, and operations. Algebraic expressions frequently involve more than one of the operation symbols that we have seen thus far in this section. For instance, when we are given an expression to evaluate such as 4 we must agree on the order in which the indicated operations are to be performed. If we don t, we can end up with different results after the evaluation. For instance, if we were to add first, in this case we would have 4 5 4 0 Add Then first multiply However, if we multiply first, we have 4 4 Multiply Then add NOTE This means that 4 4 and the second approach shown above is the correct one. Because we get different answers depending on the order in which we do the operations, the language of algebra would not be clear unless we agreed on which of the methods of evaluation shown above is correct. To avoid this difficulty, we will agree that the multiplication in an expression such as 4 should always be done before the addition. 9
0 CHAPTER THE REAL NUMBERS NOTE An algorithm is a stepby-step process for solving a problem. We refer to our procedure as the order of operations. The following algorithm gives us a set of rules, defining the order in which the operations should be performed. Rules and Properties: Order of Operations NOTE The most common grouping symbols are parentheses, brackets, fraction bars, absolute value signs, and radicals.. Simplify within the innermost grouping symbol, and work outward until all grouping symbols are removed.. Evaluate any expressions involving exponents.. Perform any multiplication and division, working from left to right. 4. Then do any addition and subtraction, again working left to right. Example Evaluating Expressions NOTE Remember: 7 7 7 49 Two factors Evaluate each expression. (a) 4 8 (b) (4 ) 7 4 (c) (4 ) (7) 49 98 Multiply first. Then do the addition. Simplify within the grouping symbol. Then multiply. Add inside the parentheses. Evaluate the power. Multiply. 8 Three factors (d) 5 5 8 40 4 40 CHECK YOURSELF Evaluate each expression. Evaluate the power. Multiply. Add and then subtract from left to right. (a) 50 6 8 (b) (5 0) (c) (5 0) (d) 7 There are several properties of the two primary operations, addition and multiplication, that are very important in the study of algebra. The following table describes several of those properties for real numbers a, b, and c. NOTE The multiplicative inverse of a is also called the reciprocal of a. This is the property that allows us to define division by any nonzero number. Property Addition Multiplication Closure a b R a b R Associative (a b) c a (b c) (a b) c a (b c) Commutative a b b a a b b a Identity a 0 a a a Inverse a ( a) 0 a a
OPERATIONS AND PROPERTIES SECTION. Example illustrates the use of the properties introduced above. Example Identifying Properties of Multiplication State the property used to justify each statement. NOTE The grouping has been changed. NOTE Because reciprocal of. is the (a) ( b) ( ) b Associative property of addition (b) Multiplicative inverse (c) ( )( 4) is a real number Closure property of multiplication (d) (5) 5 Multiplicative identity NOTE Only the order has been changed. (e) (x y) (y x) Commutative property of addition CHECK YOURSELF State the property used to justify each statement. (a) (9)( 7) is a real number (c) xy xy (e) (ab) ( )ab (b) x y y x (d) 0 x y x y In addition to the specific properties for addition and multiplication, we have one property that involves both operations. Rules and Properties: Distributive Property For any real numbers a, b, and c, a(b c) ab ac In words, multiplication distributes over addition. The following example illustrates the use of the distributive property.
CHAPTER THE REAL NUMBERS Example Using the Distributive Property NOTE Distribute the multiplication by 4 over x and 7. Simplify. Use the distributive property to simplify each expression. (a) 4(x 7) 4(x) 4(7) x 8 (b) 7(x y 5) 7(x) 7(y) 7(5) x 4y 5 CHECK YOURSELF Use the distributive property to simplify each expression. (a) 5(4a 5) (b) 4(x 5x) (c) 6(4a b 7c) (d) 5(p 5q) One of the most important uses of the distributive property relates to the combining of like terms. Example 4 illustrates. Example 4 Combining Like Terms Combine all like terms. (a) x 5x ( 5)x 7x (b) a 4b 7a b ( 7)a (4 )b 0a b CHECK YOURSELF 4 Combine all like terms. (a) x x (c) x 9x x x (b) a b 7a b CHECK YOURSELF ANSWERS. (a) ; (b) 5; (c) 75; (d) 7. (a) Closure property of multiplication; (b) commutative property of addition; (c) multiplicative inverse; (d) additive identity; (e) associative property of multiplication. (a) 0a 5; (b) 8x 0x; (c) 4a 8b 4c; (d) 5p 5q 4. (a) 5x; (b) 9a; (c) x 6x
Name. Exercises Section Date Translate each of the following statements, using symbols.. The sum of 0 and x. x plus 5. more than p 4. The sum of m and 5 5. n increased by 6. s increased by 7. m minus 4 8. 5 less than b 9. Subtract from x. 0. 5 minus a. The product of m and n. The quotient of b and. s divided by 4 4. 7 times b 5. times the difference of c and d 6. Twice the sum of a and b 7. 4 less than the product of r and s 8. more than times w ANSWERS... 4. 5. 6. 7. 8. 9. 0. 9. The sum of c and 4, divided 0. The difference of m and, by d divided by n Apply the order-of-operations algorithm to evaluate the following expressions.. 5 4 6. 7 5... 4. 5. 6. 7. 8.. 7(8 ) 4. ( 7) 5. 6(8 4) 6. 4( 6) 7. (4 )(5 ) 8. (5 6)( ) 9. 4 5 0. 5 6 9. 0.... 4. 5. 6. 7. 8.. (7 5)(7 5). ( )( ). 7 5 7 5 4. 5. 9 5 6. 7. (9 5) 8. ( ) 9. 0.... 4. 5. 6. 7. 8.
ANSWERS 9. 40. 4. 4. 4. 44. 45. 46. 47. 48. 49. 50. 9. 6 40. 0 8 4 4. 8 4 4. 48 8 4 4. ( ) 44. (4 ) 5 45. [5 (6 ) ] 46. [4 (5 ) ] 5 5 47. 48. 4 ( 8) 5 5. 5. 5. 54. 55. 56. 6 8 49. 4 50. (5)( ) 5. ( )(5) 5. 4 8 ( 8)( ) ()( 4) 57. 58. 59. 60. 6. 6. 6. 64. 65. 66. 67. 68. 69. 70. In each exercise, apply the commutative and associative properties to rewrite the expression. Then simplify the result. 5. (b 5) 54. (x ) 8 55. 8 (6 a) 56. 0 ( y) 57. (x 5) 58. (w ) 0 59. 8 ( p 6) 60. 6 (m ) 6. (8 a) ( 8) 6. (p ) 6. (8x) 64. 6(b) 7. 7. 7. 65. 66. 4 (4w) 7 7 m m 67. 68. 6p 6 4 b 4 74. 75. 76. In each exercise apply the distributive property to rewrite the expression. Then simplify the result when possible. 69. 5(m ) 70. (4p 5) 77. 78. 7. 4a(a 4) 7. 6b(b 5) 7. (4a 0) 74. (6y 5) 75. 5(a b 4) 76. 6(m 6n 7) 77. (4a 6b c) 78. (x 6y 9z) 4
ANSWERS In each exercise, apply the distributive property to simplify the expression. 79. 8b b 80. 0a a 8. m 4m m 8. b b b 8. 84. a 4 a 85. 86. a a In each exercise, apply the appropriate properties to rewrite the expression. Then simplify the result. 87. 6x ( x) 88. 5p ( 9p) 89. 8y (y 5) 90. 8m ( 4m) 9. x 9 4x 6 9. a 9a 7 9. b b 5 4b 94. 6x 7 8x 0 95. 7y ( ) y 96. w ( 7) w 7 97. (y ) y 98. 5 (b ) 4b 99. y y( y) y 00. 5n n(n ) n State the property used to justify the following statements. 0. 8 8 0. 6 is a real number 0. ( y 5) y 0 04. 6(x) (6 )x 05. 4 (5 6) (4 5) 6 06. 4 (5 6) (5 6) 4 07. 8b 6 b 8b b 6 08. 8b (b 6) (8b b) 6 09. (8b b) 6 (8 )b 6 0.. 7 7 5 5 0. ( y )( y ) y( y ) ( y ) 5 b 8 5 b 4 m 5 6 m 79. 80. 8. 8. 8. 84. 85. 86. 87. 88. 89. 90. 9. 9. 9. 94. 95. 96. 97. 98. 99. 00. 0. 0. 0. 04. 05. 06. 07. 08. 09. 0... 5
ANSWERS. 4. Determine whether each statement is true or false. If it is false, rewrite the right side of the equation to make it a true statement. 5. 6. 7. 8. 9. 0.... 4. 5.. 5( y 4) 5y 4 4. 0 5x 5 5( x 5) 5. 7b 8b 5b 6. a (0 a) (a a) 0 7. 4(w ) w 7 8. 5 y 5 y 0 y 9. m 4m 7m 0. b b 5 0b. 6y ( 6)y 0. 4b ( 4b) b. n 6n 8n 4. a a a 5. A local baker observed that the sales in her store in May were twice the sales in April. She also observed that the sales in June were three-fourths the sales in April. Use variables to describe the sales of the bakery in each of the months. 6. Computer Corner noted that the sales of computers in August were three-fourths of the sales of computers in July. The sales of computers in September were five-sixths of the sales of computers in July. Use variables to describe the number of computers sold in each of the months. 7. Create an example to show that subtraction of signed numbers is not commutative. 6. 8. Create an example to show that division of signed numbers is not associative. 7. 8. Answers. 0 x. p 5. n 7. m 4 9. x. mn. s c 4 5. (c d) 7. rs 4 9. 4 d. 9. 4 5. 96 7. 56 9.. 4. 7 5. 56 7. 6 9. 4. 4 4. 9 45. 9 47. 49. 5. 4 5. b 8 55. 4 a 57. x 7 59. p 4 6. a 6. 6x 65. w 67. 69. 0m 5 7. 4a 6a 7. a 5 75. 5a 0b 0 77. a b c 79. 0b 8. 8m 8. a 85. 5 6 a 87. 9x 89. 0y 5 9. 6x 5 9. 9b 5 95. 8y 97. 9y 5 99. 5y 9y 0. Commutative property of addition 0. Distributive property 05. Associative property of addition 07. Commutative property of addition 09. Distributive property. Additive inverse. False, 5y 0 5. True 7. False, w 9. True. True. False, 8n 5. April: x; May: x; June: 4 x 7. 6