m81316 Commutative and Associative Properties By the end of this section, you will be able to: Use the commutative and associative properties Evaluate expressions using the commutative and associative properties Simplify expressions using the commutative and associative properties 60473ca2-c628-4e39-ac2e-1383385ffd82 Before you get started, take this readiness quiz. Simplify: 7y+2+y+13. If you missed this problem, review . 8y+15 Multiply: 23·18. If you missed this problem, review . 12 Find the opposite of 15. If you missed this problem, review . −15 In the next few sections, we will take a look at the properties of real numbers. Many of these properties will describe things you already know, but it will help to give names to the properties and define them formally. This way we’ll be able to refer to them and use them as we solve equations in the next chapter.

Think about adding two numbers, such as 5 and 3. 5+33+588 The results are the same. 5+3=3+5 Notice, the order in which we add does not matter. The same is true when multiplying 5 and 3. 5·33·51515 Again, the results are the same! 5·3=3·5. The order in which we multiply does not matter. These examples illustrate the commutative properties of addition and multiplication. Commutative Property of Addition: if a and b are real numbers, then a+b=b+a Commutative Property of Multiplication: if a and b are real numbers, then a·b=b·a The commutative properties have to do with order. If you change the order of the numbers when adding or multiplying, the result is the same. Use the commutative properties to rewrite the following expressions: ⓐ −1+3=_____ ⓑ 4·9=_____ ⓐ −1+3=_____ Use the commutative property of addition to change the order. −1+3=3+(−1) ⓑ 4·9=_____ Use the commutative property of multiplication to change the order. 4·9=9·4 Use the commutative properties to rewrite the following: ⓐ −4+7=_____ⓑ 6·12=_____ ⓐ −4 + 7 = 7 + (−4)ⓑ 6 · 12 = 12 · 6 Use the commutative properties to rewrite the following: ⓐ 14+(−2)=_____ⓑ 3(−5)=_____ ⓐ 14 + (−2) = −2 + 14ⓑ 3(−5) = (−5)3 What about subtraction? Does order matter when we subtract numbers? Does 7−3 give the same result as 3−7? 7−33−74−44≠−4 The results are not the same.7−3≠3−7 Since changing the order of the subtraction did not give the same result, we can say that subtraction is not commutative. Let’s see what happens when we divide two numbers. Is division commutative? 12÷44÷121244123133≠13 The results are not the same. So12÷4≠4÷12 Since changing the order of the division did not give the same result, division is not commutative. Addition and multiplication are commutative. Subtraction and division are not commutative.Suppose you were asked to simplify this expression. 7+8+2 How would you do it and what would your answer be? Some people would think 7+8is15 and then 15+2is17. Others might start with 8+2makes10 and then 7+10makes17. Both ways give the same result, as shown in . (Remember that parentheses are grouping symbols that indicate which operations should be done first.)

When adding three numbers, changing the grouping of the numbers does not change the result. This is known as the Associative Property of Addition.The same principle holds true for multiplication as well. Suppose we want to find the value of the following expression: 5·13·3 Changing the grouping of the numbers gives the same result, as shown in .

When multiplying three numbers, changing the grouping of the numbers does not change the result. This is known as the Associative Property of Multiplication.If we multiply three numbers, changing the grouping does not affect the product.You probably know this, but the terminology may be new to you. These examples illustrate the Associative Properties. Associative Property of Addition: if a,b, and c are real numbers, then(a+b)+c=a+(b+c) Associative Property of Multiplication: if a,b, and c are real numbers, then(a·b)·c=a·(b·c) Use the associative properties to rewrite the following: ⓐ (3+0.6)+0.4=__________ ⓑ (−4·25)·15=__________ ⓐ (3+0.6)+0.4=__________ Change the grouping. (3+0.6)+0.4=3+(0.6+0.4) Notice that 0.6+0.4 is 1, so the addition will be easier if we group as shown on the right. ⓑ (−4·25)·15=__________ Change the grouping. (−4·25)·15=−4·(25·15) Notice that 25·15 is 6. The multiplication will be easier if we group as shown on the right. Use the associative properties to rewrite the following:ⓐ (1+0.7)+0.3=__________ ⓑ (−9·8)·34=__________ ⓐ (1+0.7)+0.3=1+(0.7+0.3)ⓑ (−9·8)·34=−9(8·34) Use the associative properties to rewrite the following:ⓐ (4+0.6)+0.4=__________ ⓑ (−2·12)·56=__________ ⓐ (4+0.6)+0.4=4+(0.6+0.4)ⓑ (−2·12)·56=−2(12·56) Besides using the associative properties to make calculations easier, we will often use it to simplify expressions with variables. Use the Associative Property of Multiplication to simplify: 6(3x). 6(3x) Change the grouping. (6·3)x Multiply in the parentheses. 18x Notice that we can multiply 6·3, but we could not multiply 3·x without having a value for x. Use the Associative Property of Multiplication to simplify the given expression: 8(4x). 8(4x) = (8 · 4)x = 32x Use the Associative Property of Multiplication to simplify the given expression: −9(7y). −9(7y) = (−9 · 7)y = −63y

The commutative and associative properties can make it easier to evaluate some algebraic expressions. Since order does not matter when adding or multiplying three or more terms, we can rearrange and re-group terms to make our work easier, as the next several examples illustrate. Evaluate each expression when x=78. ⓐ x+0.37+(−x) ⓑ x+(−x)+0.37 ⓐ Substitute 78 for x. Convert fractions to decimals. Add left to right. Subtract. ⓑ Substitute 78 for x. Add opposites first. What was the difference between part ⓐ and part ⓑ ? Only the order changed. By the Commutative Property of Addition, x+0.37+(−x)=x+(−x)+0.37. But wasn’t part ⓑ much easier? Evaluate each expression when y=38:ⓐ y+0.84+(−y) ⓑ y+(−y)+0.84. ⓐ 0.84ⓑ 0.84 Evaluate each expression when f=1720:ⓐ f+0.975+(−f) ⓑ f+(−f)+0.975. ⓐ 0.975ⓑ 0.975 Let’s do one more, this time with multiplication. Evaluate each expression when n=17. ⓐ 43(34n) ⓑ (43·34)n ⓐ Substitute 17 for n. Multiply in the parentheses first. Multiply again. ⓑ Substitute 17 for n. Multiply. The product of reciprocals is 1. Multiply again. What was the difference between part ⓐ and part ⓑ here? Only the grouping changed. By the Associative Property of Multiplication, 43(34n)=(43·34)n. By carefully choosing how to group the factors, we can make the work easier. Evaluate each expression when p=24:ⓐ 59(95p) ⓑ (59·95)p. ⓐ 24ⓑ 24 Evaluate each expression when q=15:ⓐ 711(117q) ⓑ (711·117)q ⓐ 15ⓑ 15

When we have to simplify algebraic expressions, we can often make the work easier by applying the Commutative or Associative Property first instead of automatically following the order of operations. Notice that in part ⓑ was easier to simplify than part ⓐ because the opposites were next to each other and their sum is 0. Likewise, part ⓑ in was easier, with the reciprocals grouped together, because their product is 1. In the next few examples, we’ll use our number sense to look for ways to apply these properties to make our work easier. Simplify: −84n+(−73n)+84n. Notice the first and third terms are opposites, so we can use the commutative property of addition to reorder the terms. −84n+(−73n)+84n Re-order the terms. −84n+84n+(−73n) Add left to right. 0+(−73n) Add. −73n Simplify: −27a+(−48a)+27a. −48a Simplify: 39x+(−92x)+(−39x). −92x Now we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals—their product is 1. Simplify: 715·823·157. Notice the first and third terms are reciprocals, so we can use the Commutative Property of Multiplication to reorder the factors. 715·823·157 Re-order the terms. 715·157·823 Multiply left to right. 1·823 Multiply. 823 Simplify: 916·549·169. 549 Simplify: 617·1125·176. 1125 In expressions where we need to add or subtract three or more fractions, combine those with a common denominator first. Simplify: (513+34)+14. Notice that the second and third terms have a common denominator, so this work will be easier if we change the grouping. (513+34)+14 Group the terms with a common denominator. 513+(34+14) Add in the parentheses first. 513+(44) Simplify the fraction. 513+1 Add. 1513 Convert to an improper fraction. 1813 Simplify: (715+58)+38. 2215 Simplify: (29+712)+512. 119 When adding and subtracting three or more terms involving decimals, look for terms that combine to give whole numbers. Simplify: (6.47q+9.99q)+1.01q. Notice that the sum of the second and third coefficients is a whole number. (6.47q+9.99q)+1.01q Change the grouping. 6.47q+(9.99q+1.01q) Add in the parentheses first. 6.47q+(11.00q) Add. 17.47q Many people have good number sense when they deal with money. Think about adding 99 cents and 1 cent. Do you see how this applies to adding 9.99+1.01? Simplify: (5.58c+8.75c)+1.25c. 15.58c Simplify: (8.79d+3.55d)+5.45d. 17.79d No matter what you are doing, it is always a good idea to think ahead. When simplifying an expression, think about what your steps will be. The next example will show you how using the Associative Property of Multiplication can make your work easier if you plan ahead. Simplify the expression: [ 1.67(8) ] (0.25). Notice that multiplying (8)(0.25) is easier than multiplying 1.67(8) because it gives a whole number. (Think about having 8 quarters—that makes $2.) [1.67(8)](0.25) Regroup. 1.67[(8)(0.25)] Multiply in the brackets first. 1.67[2] Multiply. 3.34 Simplify: [1.17(4)](2.25). 10.53 Simplify: [3.52(8)](2.5). 70.4 When simplifying expressions that contain variables, we can use the commutative and associative properties to re-order or regroup terms, as shown in the next pair of examples. Simplify: 6(9x). 6(9x) Use the associative property of multiplication to re-group. (6·9)x Multiply in the parentheses. 54x Simplify: 8(3y). 24y Simplify: 12(5z). 60z In The Language of Algebra, we learned to combine like terms by rearranging an expression so the like terms were together. We simplified the expression 3x+7+4x+5 by rewriting it as 3x+4x+7+5 and then simplified it to 7x+12. We were using the Commutative Property of Addition. Simplify: 18p+6q+(−15p)+5q. Use the Commutative Property of Addition to re-order so that like terms are together. 18p+6q+(−15p)+5q Re-order terms. 18p+(−15p)+6q+5q Combine like terms. 3p+11q Simplify: 23r+14s+9r+(−15s). 32r − s Simplify: 37m+21n+4m+(−15n). 41m + 6n The Links to Literacy activity, "Each Orange Had 8 Slices" will provide you with another view of the topics covered in this section.

Commutative PropertiesCommutative Property of Addition:If a,b are real numbers, then a+b=b+a Commutative Property of Multiplication:If a,b are real numbers, then a⋅b=b⋅a Associative PropertiesAssociative Property of Addition: If a,b,c are real numbers then (a+b)+c=a+(b+c) Associative Property of Multiplication: If a,b,c are real numbers then (a⋅b)⋅c=a⋅(b⋅c)

Use the Commutative and Associative Properties In the following exercises, use the commutative properties to rewrite the given expression. 8+9=___ 7+6=___ 7 + 6 = 6 + 7 8(−12)=___ 7(−13)=___ 7(−13) = (−13)7 (−19)(−14)=___ (−12)(−18)=___ (−12)(−18) = (−18)(−12) −11+8=___ −15+7=___ −15 + 7 = 7 + (−15) x+4=___ y+1=___ y + 1 = 1 + y −2a=___ −3m=___ −3m = m(−3) In the following exercises, use the associative properties to rewrite the given expression. (11+9)+14=___ (21+14)+9=___ (21 + 14) + 9 = 21 + (14 + 9) (12·5)·7=___ (14·6)·9=___ (14 · 6) · 9 = 14(6 · 9) (−7+9)+8=___ (−2+6)+7=___ (−2 + 6) + 7 = −2 + (6 + 7) (16·45)·15=___ (13·23)·18=___ (13·23)·18=13(23·18) 3(4x)=___ 4(7x)=___ 4(7x) = (4 · 7)x (12+x)+28=___ (17+y)+33=___ (17 + y) + 33 = 17 + (y + 33) Evaluate Expressions using the Commutative and Associative Properties In the following exercises, evaluate each expression for the given value. If y=58, evaluate: ⓐ y+0.49+(−y)ⓑ y+(−y)+0.49 If z=78, evaluate: ⓐ z+0.97+(−z)ⓑ z+(−z)+0.97 ⓐ 0.97 ⓑ 0.97 If c=−114, evaluate: ⓐ c+3.125+(−c)ⓑ c+(−c)+3.125 If d=−94, evaluate: ⓐ d+2.375+(−d)ⓑ d+(−d)+2.375 ⓐ 2.375ⓑ 2.375 If j=11, evaluate: ⓐ 56(65j)ⓑ (56·65)j If k=21, evaluate: ⓐ 413(134k)ⓑ (413·134)k ⓐ 21ⓑ 21 If m=−25, evaluate: ⓐ −37(73m)ⓑ (−37·73)m If n=−8, evaluate: ⓐ −521(215n)ⓑ (−521·215)n ⓐ 8ⓑ 8 Simplify Expressions Using the Commutative and Associative Properties In the following exercises, simplify. −45a+15+45a 9y+23+(−9y) 23 12+78+(−12) 25+512+(−25) 512 320·4911·203 1318·257·1813 257 712·917·247 310·1323·503 6523 −24·7·38 −36·11·49 −176 (56+815)+715 (112+49)+59 1312 513+34+14 815+57+27 2315 (4.33p+1.09p)+3.91p (5.89d+2.75d)+1.25d 9.89d 17(0.25)(4) 36(0.2)(5) 36 [2.48(12)](0.5) [9.731(4)](0.75) 29.193 7(4a) 9(8w) 72w −15(5m) −23(2n) −46n 12(56p) 20(35q) 12q 14x+19y+25x+3y 15u+11v+27u+19v 42u + 30v 43m+(−12n)+​(−16m)+(−9n) −22p+17q+(−35p)+(−27q) −57p + (−10q) 38g+112h+78g+512h 56a+310b+16a+910b a+65b 6.8p+9.14q+(−4.37p)+(−0.88q) 9.6m+7.22n+(−2.19m)+(−0.65n) 7.41m + 6.57n

Stamps Allie and Loren need to buy stamps. Allie needs four $0.49 stamps and nine $0.02 stamps. Loren needs eight $0.49 stamps and three $0.02 stamps. ⓐ How much will Allie’s stamps cost? ⓑ How much will Loren’s stamps cost? ⓒ What is the total cost of the girls’ stamps? ⓓ How many $0.49 stamps do the girls need altogether? How much will they cost? ⓔ How many $0.02 stamps do the girls need altogether? How much will they cost? Counting Cash Grant is totaling up the cash from a fundraising dinner. In one envelope, he has twenty-three $5 bills, eighteen $10 bills, and thirty-four $20 bills. In another envelope, he has fourteen $5 bills, nine $10 bills, and twenty-seven $20 bills. ⓐ How much money is in the first envelope? ⓑ How much money is in the second envelope? ⓒ What is the total value of all the cash? ⓓ What is the value of all the $5 bills? ⓔ What is the value of all $10 bills? ⓕ What is the value of all $20 bills? ⓐ $975ⓑ $700ⓒ $1675ⓓ $185ⓔ $270ⓕ $1220

In your own words, state the Commutative Property of Addition and explain why it is useful. In your own words, state the Associative Property of Multiplication and explain why it is useful. Answers will vary.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?