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== 7.2#h(0.6em)Commutative and Associative Properties

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Before you get started, take this readiness quiz.

Simplify: #math.equation(block: false, alt: "7 y plus 2 plus y plus 13 .")[$7 y + 2 + y + 13 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "8 y plus 15")[$8 y + 15$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "the fraction 2 over 3 times 18 .")[$frac(2, 3) · 18 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "12")[$12$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the opposite of #math.equation(block: false, alt: "15 .")[$15 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "minus 15")[$− 15$]
]
]

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.


=== Use the Commutative and Associative Properties

Think about adding two numbers, such as #math.equation(block: false, alt: "5")[$5$] and #math.equation(block: false, alt: "3 .")[$3 .$]

#math.equation(block: true, alt: "5 plus 3, 3 plus 5; 8, 8")[$5 + 3 &  &  & 3 + 5 \ 8 &  &  & 8$]The results are the same. #math.equation(block: false, alt: "5 plus 3 equals 3 plus 5")[$5 + 3 = 3 + 5$]

Notice, the order in which we add does not matter. The same is true when multiplying #math.equation(block: false, alt: "5")[$5$] and #math.equation(block: false, alt: "3 .")[$3 .$]

#math.equation(block: true, alt: "5 times 3, 3 times 5; 15, 15")[$5 · 3 &  &  & 3 · 5 \ 15 &  &  & 15$]Again, the results are the same! #math.equation(block: false, alt: "5 times 3 equals 3 times 5 .")[$5 · 3 = 3 · 5 .$] The order in which we multiply does not matter.

These examples illustrate the commutative properties of addition and multiplication.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Commutative Properties]

#strong[Commutative Property of Addition]: if #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] are real numbers, then

#math.equation(block: true, alt: "a plus b equals b plus a")[$a + b = b + a$]#strong[Commutative Property of Multiplication]: if #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] are real numbers, then

#math.equation(block: true, alt: "a times b equals b times a")[$a · b = b · a$]
]

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.

#examplebox("Example 1")[][
Use the commutative properties to rewrite the following expressions:

+ ⓐ #math.equation(block: false, alt: "−1 plus 3 equals _____")[$#h(0.2em) −1 + 3 = \_\_\_\_\_$]
+ ⓑ #math.equation(block: false, alt: "4 times 9 equals _____")[$#h(0.2em) 4 · 9 = \_\_\_\_\_$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "−1 plus 3 equals _____")[$−1 + 3 = \_\_\_\_\_$]],
  [Use the commutative property of addition to change the order.], [#math.equation(block: false, alt: "−1 plus 3 equals 3 plus open parenthesis −1 close parenthesis")[$−1 + 3 = 3 + ( −1 )$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "4 times 9 equals _____")[$4 · 9 = \_\_\_\_\_$]],
  [Use the commutative property of multiplication to change the order.], [#math.equation(block: false, alt: "4 times 9 equals 9 times 4")[$4 · 9 = 9 · 4$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the commutative properties to rewrite the following:

+ ⓐ #math.equation(block: false, alt: "−4 plus 7 equals _____")[$#h(0.2em) −4 + 7 = \_\_\_\_\_ #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "6 times 12 equals _____")[$#h(0.2em) 6 · 12 = \_\_\_\_\_$]

#solutionbox[
+ ⓐ −4 + 7 = 7 + (−4)
+ ⓑ 6 · 12 = 12 · 6
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the commutative properties to rewrite the following:

+ ⓐ #math.equation(block: false, alt: "14 plus open parenthesis −2 close parenthesis equals _____")[$#h(0.2em) 14 + ( −2 ) = \_\_\_\_\_ #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "3 open parenthesis −5 close parenthesis equals _____")[$#h(0.2em) 3 ( −5 ) = \_\_\_\_\_$]

#solutionbox[
+ ⓐ 14 + (−2) = −2 + 14
+ ⓑ 3(−5) = (−5)3
]
]

What about subtraction? Does order matter when we subtract numbers? Does #math.equation(block: false, alt: "7 minus 3")[$7 − 3$] give the same result as #math.equation(block: false, alt: "3 minus 7 ?")[$3 − 7 ?$]

#math.equation(block: true, alt: "7 minus 3, 3 minus 7; 4, minus 4; 4 not equal to −4")[$7 − 3 &  & 3 − 7 \ 4 &  & − 4 \  & 4 ≠ −4 &$]#math.equation(block: true, alt: "The results are not the same. 7 minus 3 not equal to 3 minus 7")[$"The results are not the same." #h(0.2em) 7 − 3 ≠ 3 − 7$]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?

#math.equation(block: true, alt: "12 divided by 4, 4 divided by 12; the fraction 12 over 4, the fraction 4 over 12; 3, the fraction 1 over 3; 3 not equal to the fraction 1 over 3")[$12 ÷ 4 &  & 4 ÷ 12 \ frac(12, 4) &  & frac(4, 12) \ 3 &  & frac(1, 3) \  & 3 ≠ frac(1, 3) &$]#math.equation(block: true, alt: "The results are not the same. So 12 divided by 4 not equal to 4 divided by 12")[$"The results are not the same. So" #h(0.2em) 12 ÷ 4 ≠ 4 ÷ 12$]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.

#math.equation(block: true, alt: "7 plus 8 plus 2")[$7 + 8 + 2$]How would you do it and what would your answer be?

Some people would think #math.equation(block: false, alt: "7 plus 8 is 15")[$7 + 8 #h(0.2em) "is" #h(0.2em) 15$] and then #math.equation(block: false, alt: "15 plus 2 is 17 .")[$15 + 2 #h(0.2em) "is" #h(0.2em) 17 .$] Others might start with #math.equation(block: false, alt: "8 plus 2 makes 10")[$8 + 2 #h(0.2em) "makes" #h(0.2em) 10$] and then #math.equation(block: false, alt: "7 plus 10 makes 17 .")[$7 + 10 #h(0.2em) "makes" #h(0.2em) 17 .$]

Both ways give the same result, as shown. (Remember that parentheses are grouping symbols that indicate which operations should be done first.)

#figure(figph[The image shows an equation. The left side of the equation shows the quantity 7 plus 8 in parentheses plus 2. The right side of the equation show 7 plus the quantity 8 plus 2. Each side of the equation is boxed separately in red. Each box has an arrow pointing from the box to the number 17 below.], alt: "The image shows an equation. The left side of the equation shows the quantity 7 plus 8 in parentheses plus 2. The right side of the equation show 7 plus the quantity 8 plus 2. Each side of the equation is boxed separately in red. Each box has an arrow pointing from the box to the number 17 below.", caption: none)

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:

#math.equation(block: true, alt: "5 times the fraction 1 over 3 times 3")[$5 · frac(1, 3) · 3$]Changing the grouping of the numbers gives the same result, as shown.

#figure(figph[The image shows an equation. The left side of the equation shows the quantity 5 times 1 third in parentheses times 3. The right side of the equation show 5 times the quantity 1 third times 3. Each side of the equation is boxed separately in red. Each box has an arrow pointing from the box to the number 5 below.], alt: "The image shows an equation. The left side of the equation shows the quantity 5 times 1 third in parentheses times 3. The right side of the equation show 5 times the quantity 1 third times 3. Each side of the equation is boxed separately in red. Each box has an arrow pointing from the box to the number 5 below.", caption: none)

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 #emph[Associative Properties].

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Associative Properties]

#emph[Associative Property of Addition]: if #math.equation(block: false, alt: "a , b ,")[$a , b ,$] and #math.equation(block: false, alt: "c")[$c$] are real numbers, then

#math.equation(block: true, alt: "open parenthesis a plus b close parenthesis plus c equals a plus open parenthesis b plus c close parenthesis")[$( a + b ) + c = a + ( b + c )$]#emph[Associative Property of Multiplication]: if #math.equation(block: false, alt: "a , b ,")[$a , b ,$] and #math.equation(block: false, alt: "c")[$c$] are real numbers, then

#math.equation(block: true, alt: "open parenthesis a times b close parenthesis times c equals a times open parenthesis b times c close parenthesis")[$( a · b ) · c = a · ( b · c )$]
]

#examplebox("Example 2")[][
Use the associative properties to rewrite the following:

+ ⓐ #math.equation(block: false, alt: "open parenthesis 3 plus 0.6 close parenthesis plus 0.4 equals __________")[$#h(0.2em) ( 3 + 0.6 ) + 0.4 = \_\_\_\_\_\_\_\_\_\_$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis −4 times the fraction 2 over 5 close parenthesis times 15 equals __________")[$#h(0.2em) ( −4 · frac(2, 5) ) · 15 = \_\_\_\_\_\_\_\_\_\_$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "open parenthesis 3 plus 0.6 close parenthesis plus 0.4 equals __________")[$( 3 + 0.6 ) + 0.4 = \_\_\_\_\_\_\_\_\_\_$]],
  [Change the grouping.], [#math.equation(block: false, alt: "open parenthesis 3 plus 0.6 close parenthesis plus 0.4 equals 3 plus open parenthesis 0.6 plus 0.4 close parenthesis")[$( 3 + 0.6 ) + 0.4 = 3 + ( 0.6 + 0.4 )$]],
))

Notice that #math.equation(block: false, alt: "0.6 plus 0.4")[$0.6 + 0.4$] is #math.equation(block: false, alt: "1 ,")[$1 ,$] so the addition will be easier if we group as shown on the right.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "open parenthesis −4 times the fraction 2 over 5 close parenthesis times 15 equals __________")[$( −4 · frac(2, 5) ) · 15 = \_\_\_\_\_\_\_\_\_\_$]],
  [Change the grouping.], [#math.equation(block: false, alt: "open parenthesis −4 times the fraction 2 over 5 close parenthesis times 15 equals −4 times open parenthesis the fraction 2 over 5 times 15 close parenthesis")[$( −4 · frac(2, 5) ) · 15 = −4 · ( frac(2, 5) · 15 )$]],
))

Notice that #math.equation(block: false, alt: "the fraction 2 over 5 times 15")[$frac(2, 5) · 15$] is #math.equation(block: false, alt: "6 .")[$6 .$] The multiplication will be easier if we group as shown on the right.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the associative properties to rewrite the following: #linebreak() ⓐ  #math.equation(block: false, alt: "open parenthesis 1 plus 0.7 close parenthesis plus 0.3 equals __________")[$( 1 + 0.7 ) + 0.3 = \_\_\_\_\_\_\_\_\_\_ #h(2em)$] ⓑ #math.equation(block: false, alt: "open parenthesis −9 times 8 close parenthesis times the fraction 3 over 4 equals __________")[$#h(0.2em) ( −9 · 8 ) · frac(3, 4) = \_\_\_\_\_\_\_\_\_\_$]

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "open parenthesis 1 plus 0.7 close parenthesis plus 0.3 equals 1 plus open parenthesis 0.7 plus 0.3 close parenthesis")[$#h(0.2em) ( 1 + 0.7 ) + 0.3 = 1 + ( 0.7 + 0.3 )$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis −9 times 8 close parenthesis times the fraction 3 over 4 equals −9 open parenthesis 8 times the fraction 3 over 4 close parenthesis")[$#h(0.2em) ( −9 · 8 ) · frac(3, 4) = −9 ( 8 · frac(3, 4) )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the associative properties to rewrite the following: #linebreak() ⓐ #math.equation(block: false, alt: "open parenthesis 4 plus 0.6 close parenthesis plus 0.4 equals __________")[$( 4 + 0.6 ) + 0.4 = \_\_\_\_\_\_\_\_\_\_ #h(2em)$] ⓑ #math.equation(block: false, alt: "open parenthesis −2 times 12 close parenthesis times the fraction 5 over 6 equals __________")[$#h(0.2em) ( −2 · 12 ) · frac(5, 6) = \_\_\_\_\_\_\_\_\_\_$]

#solutionbox[
+ ⓐ #math.equation(block: false, alt: "open parenthesis 4 plus 0.6 close parenthesis plus 0.4 equals 4 plus open parenthesis 0.6 plus 0.4 close parenthesis")[$#h(0.2em) ( 4 + 0.6 ) + 0.4 = 4 + ( 0.6 + 0.4 )$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis −2 times 12 close parenthesis times the fraction 5 over 6 equals −2 open parenthesis 12 times the fraction 5 over 6 close parenthesis")[$#h(0.2em) ( −2 · 12 ) · frac(5, 6) = −2 ( 12 · frac(5, 6) )$]
]
]

Besides using the associative properties to make calculations easier, we will often use it to simplify expressions with variables.

#examplebox("Example 3")[][
Use the Associative Property of Multiplication to simplify: #math.equation(block: false, alt: "6 open parenthesis 3 x close parenthesis .")[$6 ( 3 x ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "6 open parenthesis 3 x close parenthesis")[$6 ( 3 x )$]]),
  [Change the grouping.], [#math.equation(block: false, alt: "open parenthesis 6 times 3 close parenthesis x")[$( 6 · 3 ) x$]],
  [Multiply in the parentheses.], [#math.equation(block: false, alt: "18 x")[$18 x$]],
))

Notice that we can multiply #math.equation(block: false, alt: "6 times 3 ,")[$6 · 3 ,$] but we could not multiply #math.equation(block: false, alt: "3 times x")[$3 · x$] without having a value for #math.equation(block: false, alt: "x .")[$x .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the Associative Property of Multiplication to simplify the given expression: #math.equation(block: false, alt: "8 open parenthesis 4 x close parenthesis .")[$8 ( 4 x ) .$]

#solutionbox[
8(4#emph[x]) = (8 · 4)#emph[x] = 32#emph[x]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the Associative Property of Multiplication to simplify the given expression: #math.equation(block: false, alt: "−9 open parenthesis 7 y close parenthesis .")[$−9 ( 7 y ) .$]

#solutionbox[
−9(7#emph[y]) = (−9 · 7)#emph[y] = −63#emph[y]
]
]


=== Evaluate Expressions using the Commutative and Associative Properties

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.

#examplebox("Example 4")[][
Evaluate each expression when #math.equation(block: false, alt: "x equals the fraction 7 over 8 .")[$x = frac(7, 8) .$]

+ ⓐ #math.equation(block: false, alt: "x plus 0.37 plus open parenthesis minus x close parenthesis")[$#h(0.2em) x + 0.37 + ( − x )$]
+ ⓑ #math.equation(block: false, alt: "x plus open parenthesis minus x close parenthesis plus 0.37")[$#h(0.2em) x + ( − x ) + 0.37$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#figure(figph[The expression x + 0.37 + (-x) illustrates the concept of additive inverses, where x and -x cancel out, leaving 0.37.], alt: "The expression x + 0.37 + (-x) illustrates the concept of additive inverses, where x and -x cancel out, leaving 0.37.", caption: none)],
  [Substitute #math.equation(block: false, alt: "the fraction 7 over 8")[$frac(7, 8)$] for #math.equation(block: false, alt: "x")[$x$].], [#figure(figph[The equation 7/8 + 0.37 + (-7/8) is displayed, where 7/8 and -7/8 are additive inverses, simplifying the sum to 0.37.], alt: "The equation 7/8 + 0.37 + (-7/8) is displayed, where 7/8 and -7/8 are additive inverses, simplifying the sum to 0.37.", caption: none)],
  [Convert fractions to decimals.], [#figure(figph[The image shows the mathematical expression: 0.875 + 0.37 + (-0.875).], alt: "The image shows the mathematical expression: 0.875 + 0.37 + (-0.875).", caption: none)],
  [Add left to right.], [#figure(figph[The image displays a subtraction problem with decimal numbers: 1.245 - 0.875, presented in a clean, straightforward manner against a white background.], alt: "The image displays a subtraction problem with decimal numbers: 1.245 - 0.875, presented in a clean, straightforward manner against a white background.", caption: none)],
  [Subtract.], [#figure(figph[The number 0.37 is displayed in black text against a plain white background.], alt: "The number 0.37 is displayed in black text against a plain white background.", caption: none)],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#figure(figph[A mathematical expression displays x plus the quantity negative x, plus 0.37, illustrating the additive inverse property where x and -x sum to zero, leaving only 0.37.], alt: "A mathematical expression displays x plus the quantity negative x, plus 0.37, illustrating the additive inverse property where x and -x sum to zero, leaving only 0.37.", caption: none)],
  [Substitute #math.equation(block: false, alt: "the fraction 7 over 8")[$frac(7, 8)$] for x.], [#figure(figph[A mathematical expression showing the sum of 7/8, its additive inverse -7/8, and the decimal 0.37. The fractions are displayed in red text, while the operations and the decimal are in black.], alt: "A mathematical expression showing the sum of 7/8, its additive inverse -7/8, and the decimal 0.37. The fractions are displayed in red text, while the operations and the decimal are in black.", caption: none)],
  [Add opposites first.], [#figure(figph[The number 0.37 is displayed in black text on a white background.], alt: "The number 0.37 is displayed in black text on a white background.", caption: none)],
))

What was the difference between part ⓐ and part ⓑ ? Only the order changed. By the Commutative Property of Addition, #math.equation(block: false, alt: "x plus 0.37 plus open parenthesis minus x close parenthesis equals x plus open parenthesis minus x close parenthesis plus 0.37 .")[$x + 0.37 + ( − x ) = x + ( − x ) + 0.37 .$] But wasn’t part ⓑ much easier?
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate each expression when #math.equation(block: false, alt: "y equals the fraction 3 over 8 :")[$y = frac(3, 8) : #h(0.2em)$]ⓐ #math.equation(block: false, alt: "y plus 0.84 plus open parenthesis minus y close parenthesis")[$#h(0.2em) y + 0.84 + ( − y )$] ⓑ #math.equation(block: false, alt: "y plus open parenthesis minus y close parenthesis plus 0.84 .")[$#h(0.2em) y + ( − y ) + 0.84 .$]

#solutionbox[
+ ⓐ 0.84
+ ⓑ 0.84
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate each expression when #math.equation(block: false, alt: "f equals the fraction 17 over 20 :")[$f = frac(17, 20) : #h(0.2em)$]ⓐ #math.equation(block: false, alt: "f plus 0.975 plus open parenthesis minus f close parenthesis")[$#h(0.2em) f + 0.975 + ( − f )$] ⓑ  #math.equation(block: false, alt: "f plus open parenthesis minus f close parenthesis plus 0.975 .")[$#h(0.2em) f + ( − f ) + 0.975 .$]

#solutionbox[
+ ⓐ 0.975
+ ⓑ 0.975
]
]

Let’s do one more, this time with multiplication.

#examplebox("Example 5")[][
Evaluate each expression when #math.equation(block: false, alt: "n equals 17 .")[$n = 17 .$]

+ ⓐ #math.equation(block: false, alt: "the fraction 4 over 3 open parenthesis the fraction 3 over 4 n close parenthesis")[$#h(0.2em) frac(4, 3) ( frac(3, 4) n )$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis the fraction 4 over 3 times the fraction 3 over 4 close parenthesis n")[$#h(0.2em) ( frac(4, 3) · frac(3, 4) ) n$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#figure(figph[A mathematical expression shows the fraction 4/3 multiplied by the quantity (3/4 multiplied by n), enclosed in parentheses.], alt: "A mathematical expression shows the fraction 4/3 multiplied by the quantity (3/4 multiplied by n), enclosed in parentheses.", caption: none)],
  [Substitute 17 for n.], [#figure(figph[A mathematical expression showing the fraction 4/3 multiplied by a parenthesized term (3/4 multiplied by the number 17, which is highlighted in red).], alt: "A mathematical expression showing the fraction 4/3 multiplied by a parenthesized term (3/4 multiplied by the number 17, which is highlighted in red).", caption: none)],
  [Multiply in the parentheses first.], [#figure(figph[A mathematical expression showing the fraction 4/3 multiplied by the fraction 51/4, enclosed in parentheses, against a white background.], alt: "A mathematical expression showing the fraction 4/3 multiplied by the fraction 51/4, enclosed in parentheses, against a white background.", caption: none)],
  [Multiply again.], [#figure(figph[The number 17 is displayed in black sans-serif font against a plain white background.], alt: "The number 17 is displayed in black sans-serif font against a plain white background.", caption: none)],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#figure(figph[The image shows a mathematical expression: an open parenthesis, the fraction 4/3, a multiplication dot, the fraction 3/4, a close parenthesis, and the variable 'n'. The expression is (4/3 \* 3/4)n.], alt: "The image shows a mathematical expression: an open parenthesis, the fraction 4/3, a multiplication dot, the fraction 3/4, a close parenthesis, and the variable 'n'. The expression is (4/3 * 3/4)n.", caption: none)],
  [Substitute 17 for n.], [#figure(figph[A mathematical expression showing the product of two fractions, (4/3) and (3/4), inside parentheses, which is then multiplied by 17, with 17 highlighted in red.], alt: "A mathematical expression showing the product of two fractions, (4/3) and (3/4), inside parentheses, which is then multiplied by 17, with 17 highlighted in red.", caption: none)],
  [Multiply. The product of reciprocals is 1.], [#figure(figph[(1) \* 17 is displayed on a white background.], alt: "(1) * 17 is displayed on a white background.", caption: none)],
  [Multiply again.], [#figure(figph[The number '17' is displayed.], alt: "The number '17' is displayed.", caption: none)],
))

What was the difference between part ⓐ and part ⓑ here? Only the grouping changed. By the Associative Property of Multiplication, #math.equation(block: false, alt: "the fraction 4 over 3 open parenthesis the fraction 3 over 4 n close parenthesis equals open parenthesis the fraction 4 over 3 times the fraction 3 over 4 close parenthesis n .")[$frac(4, 3) ( frac(3, 4) n ) = ( frac(4, 3) · frac(3, 4) ) n .$] By carefully choosing how to group the factors, we can make the work easier.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate each expression when #math.equation(block: false, alt: "p equals 24 :")[$p = 24 ":" #h(0.2em)$]ⓐ #math.equation(block: false, alt: "the fraction 5 over 9 open parenthesis the fraction 9 over 5 p close parenthesis")[$#h(0.2em) frac(5, 9) ( frac(9, 5) p ) #h(2em)$]

ⓑ #math.equation(block: false, alt: "open parenthesis the fraction 5 over 9 times the fraction 9 over 5 close parenthesis p .")[$#h(0.2em) ( frac(5, 9) · frac(9, 5) ) p .$]

#solutionbox[
+ ⓐ 24
+ ⓑ 24
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate each expression when #math.equation(block: false, alt: "q equals 15 :")[$q = 15 ":" #h(0.2em)$]ⓐ #math.equation(block: false, alt: "the fraction 7 over 11 open parenthesis the fraction 11 over 7 q close parenthesis")[$#h(0.2em) frac(7, 11) ( frac(11, 7) q ) #h(2em)$]

ⓑ #math.equation(block: false, alt: "open parenthesis the fraction 7 over 11 times the fraction 11 over 7 close parenthesis q")[$#h(0.2em) ( frac(7, 11) · frac(11, 7) ) q$]

#solutionbox[
+ ⓐ 15
+ ⓑ 15
]
]


=== Simplify Expressions Using the Commutative and Associative Properties

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 #math.equation(block: false, alt: "0 .")[$0 .$] Likewise, part ⓑ in  was easier, with the reciprocals grouped together, because their product is #math.equation(block: false, alt: "1 .")[$1 .$] In the next few examples, we’ll use our number sense to look for ways to apply these properties to make our work easier.

#examplebox("Example 6")[][
Simplify: #math.equation(block: false, alt: "−84 n plus open parenthesis −73 n close parenthesis plus 84 n .")[$−84 n + ( −73 n ) + 84 n .$]

#solutionbox[
Notice the first and third terms are opposites, so we can use the commutative property of addition to reorder the terms.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "−84 n plus open parenthesis −73 n close parenthesis plus 84 n")[$−84 n + ( −73 n ) + 84 n$]]),
  [Re-order the terms.], [#math.equation(block: false, alt: "−84 n plus 84 n plus open parenthesis −73 n close parenthesis")[$−84 n + 84 n + ( −73 n )$]],
  [Add left to right.], [#math.equation(block: false, alt: "0 plus open parenthesis −73 n close parenthesis")[$0 + ( −73 n )$]],
  [Add.], [#math.equation(block: false, alt: "−73 n")[$−73 n$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "−27 a plus open parenthesis −48 a close parenthesis plus 27 a .")[$−27 a + ( −48 a ) + 27 a .$]

#solutionbox[
−48#emph[a]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "39 x plus open parenthesis −92 x close parenthesis plus open parenthesis −39 x close parenthesis .")[$39 x + ( −92 x ) + ( −39 x ) .$]

#solutionbox[
−92#emph[x]
]
]

Now we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals—their product is #math.equation(block: false, alt: "1 .")[$1 .$]

#examplebox("Example 7")[][
Simplify: #math.equation(block: false, alt: "the fraction 7 over 15 times the fraction 8 over 23 times the fraction 15 over 7 .")[$frac(7, 15) · frac(8, 23) · frac(15, 7) .$]

#solutionbox[
Notice the first and third terms are reciprocals, so we can use the Commutative Property of Multiplication to reorder the factors.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 7 over 15 times the fraction 8 over 23 times the fraction 15 over 7")[$frac(7, 15) · frac(8, 23) · frac(15, 7)$]]),
  [Re-order the terms.], [#math.equation(block: false, alt: "the fraction 7 over 15 times the fraction 15 over 7 times the fraction 8 over 23")[$frac(7, 15) · frac(15, 7) · frac(8, 23)$]],
  [Multiply left to right.], [#math.equation(block: false, alt: "1 times the fraction 8 over 23")[$1 · frac(8, 23)$]],
  [Multiply.], [#math.equation(block: false, alt: "the fraction 8 over 23")[$frac(8, 23)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 9 over 16 times the fraction 5 over 49 times the fraction 16 over 9 .")[$frac(9, 16) · frac(5, 49) · frac(16, 9) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 over 49")[$frac(5, 49)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 6 over 17 times the fraction 11 over 25 times the fraction 17 over 6 .")[$frac(6, 17) · frac(11, 25) · frac(17, 6) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 11 over 25")[$frac(11, 25)$]
]
]

In expressions where we need to add or subtract three or more fractions, combine those with a common denominator first.

#examplebox("Example 8")[][
Simplify: #math.equation(block: false, alt: "open parenthesis the fraction 5 over 13 plus the fraction 3 over 4 close parenthesis plus the fraction 1 over 4 .")[$( frac(5, 13) + frac(3, 4) ) + frac(1, 4) .$]

#solutionbox[
Notice that the second and third terms have a common denominator, so this work will be easier if we change the grouping.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis the fraction 5 over 13 plus the fraction 3 over 4 close parenthesis plus the fraction 1 over 4")[$( frac(5, 13) + frac(3, 4) ) + frac(1, 4)$]]),
  [Group the terms with a common denominator.], [#math.equation(block: false, alt: "the fraction 5 over 13 plus open parenthesis the fraction 3 over 4 plus the fraction 1 over 4 close parenthesis")[$frac(5, 13) + ( frac(3, 4) + frac(1, 4) )$]],
  [Add in the parentheses first.], [#math.equation(block: false, alt: "the fraction 5 over 13 plus open parenthesis the fraction 4 over 4 close parenthesis")[$frac(5, 13) + ( frac(4, 4) )$]],
  [Simplify the fraction.], [#math.equation(block: false, alt: "the fraction 5 over 13 plus 1")[$frac(5, 13) + 1$]],
  [Add.], [#math.equation(block: false, alt: "1 the fraction 5 over 13")[$1 frac(5, 13)$]],
  [Convert to an improper fraction.], [#math.equation(block: false, alt: "the fraction 18 over 13")[$frac(18, 13)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis the fraction 7 over 15 plus the fraction 5 over 8 close parenthesis plus the fraction 3 over 8 .")[$( frac(7, 15) + frac(5, 8) ) + frac(3, 8) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 22 over 15")[$frac(22, 15)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis the fraction 2 over 9 plus the fraction 7 over 12 close parenthesis plus the fraction 5 over 12 .")[$( frac(2, 9) + frac(7, 12) ) + frac(5, 12) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 11 over 9")[$frac(11, 9)$]
]
]

When adding and subtracting three or more terms involving decimals, look for terms that combine to give whole numbers.

#examplebox("Example 9")[][
Simplify: #math.equation(block: false, alt: "open parenthesis 6.47 q plus 9.99 q close parenthesis plus 1.01 q .")[$( 6.47 q + 9.99 q ) + 1.01 q .$]

#solutionbox[
Notice that the sum of the second and third coefficients is a whole number.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 6.47 q plus 9.99 q close parenthesis plus 1.01 q")[$( 6.47 q + 9.99 q ) + 1.01 q$]]),
  [Change the grouping.], [#math.equation(block: false, alt: "6.47 q plus open parenthesis 9.99 q plus 1.01 q close parenthesis")[$6.47 q + ( 9.99 q + 1.01 q )$]],
  [Add in the parentheses first.], [#math.equation(block: false, alt: "6.47 q plus open parenthesis 11.00 q close parenthesis")[$6.47 q + ( 11.00 q )$]],
  [Add.], [#math.equation(block: false, alt: "17.47 q")[$17.47 q$]],
))

Many people have good number sense when they deal with money. Think about adding #math.equation(block: false, alt: "99")[$99$] cents and #math.equation(block: false, alt: "1")[$1$] cent. Do you see how this applies to adding #math.equation(block: false, alt: "9.99 plus 1.01 ?")[$9.99 + 1.01 ?$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis 5.58 c plus 8.75 c close parenthesis plus 1.25 c .")[$( 5.58 c + 8.75 c ) + 1.25 c .$]

#solutionbox[
15.58#emph[c]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis 8.79 d plus 3.55 d close parenthesis plus 5.45 d .")[$( 8.79 d + 3.55 d ) + 5.45 d .$]

#solutionbox[
17.79#emph[d]
]
]

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.

#examplebox("Example 10")[][
Simplify the expression: #math.equation(block: false, alt: "[ 1.67 open parenthesis 8 close parenthesis ] open parenthesis 0.25 close parenthesis .")[$[ 1.67 ( 8 ) ] ( 0.25 ) .$]

#solutionbox[
Notice that multiplying #math.equation(block: false, alt: "open parenthesis 8 close parenthesis open parenthesis 0.25 close parenthesis")[$( 8 ) ( 0.25 )$] is easier than multiplying #math.equation(block: false, alt: "1.67 open parenthesis 8 close parenthesis")[$1.67 ( 8 )$] because it gives a whole number. (Think about having #math.equation(block: false, alt: "8")[$8$] quarters—that makes #math.equation(block: false, alt: "$2.)")[$"$2.)"$]

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "[ 1.67 open parenthesis 8 close parenthesis ] open parenthesis 0.25 close parenthesis")[$[ 1.67 ( 8 ) ] ( 0.25 )$]]),
  [Regroup.], [#math.equation(block: false, alt: "1.67 [ open parenthesis 8 close parenthesis open parenthesis 0.25 close parenthesis ]")[$1.67 [ ( 8 ) ( 0.25 ) ]$]],
  [Multiply in the brackets first.], [#math.equation(block: false, alt: "1.67 [ 2 ]")[$1.67 [ 2 ]$]],
  [Multiply.], [#math.equation(block: false, alt: "3.34")[$3.34$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "[ 1.17 open parenthesis 4 close parenthesis ] open parenthesis 2.25 close parenthesis .")[$[ 1.17 ( 4 ) ] ( 2.25 ) .$]

#solutionbox[
10.53
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "[ 3.52 open parenthesis 8 close parenthesis ] open parenthesis 2.5 close parenthesis .")[$[ 3.52 ( 8 ) ] ( 2.5 ) .$]

#solutionbox[
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.

#examplebox("Example 11")[][
Simplify: #math.equation(block: false, alt: "6 open parenthesis 9 x close parenthesis .")[$6 ( 9 x ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "6 open parenthesis 9 x close parenthesis")[$6 ( 9 x )$]]),
  [Use the associative property of multiplication to re-group.], [#math.equation(block: false, alt: "open parenthesis 6 times 9 close parenthesis x")[$( 6 · 9 ) x$]],
  [Multiply in the parentheses.], [#math.equation(block: false, alt: "54 x")[$54 x$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "8 open parenthesis 3 y close parenthesis .")[$8 ( 3 y ) .$]

#solutionbox[
24#emph[y]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "12 open parenthesis 5 z close parenthesis .")[$12 ( 5 z ) .$]

#solutionbox[
60#emph[z]
]
]

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 #math.equation(block: false, alt: "3 x plus 7 plus 4 x plus 5")[$3 x + 7 + 4 x + 5$] by rewriting it as #math.equation(block: false, alt: "3 x plus 4 x plus 7 plus 5")[$3 x + 4 x + 7 + 5$] and then simplified it to #math.equation(block: false, alt: "7 x plus 12 .")[$7 x + 12 .$] We were using the Commutative Property of Addition.

#examplebox("Example 12")[][
Simplify: #math.equation(block: false, alt: "18 p plus 6 q plus open parenthesis −15 p close parenthesis plus 5 q .")[$18 p + 6 q + ( −15 p ) + 5 q .$]

#solutionbox[
Use the Commutative Property of Addition to re-order so that like terms are together.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "18 p plus 6 q plus open parenthesis −15 p close parenthesis plus 5 q")[$18 p + 6 q + ( −15 p ) + 5 q$]]),
  [Re-order terms.], [#math.equation(block: false, alt: "18 p plus open parenthesis −15 p close parenthesis plus 6 q plus 5 q")[$18 p + ( −15 p ) + 6 q + 5 q$]],
  [Combine like terms.], [#math.equation(block: false, alt: "3 p plus 11 q")[$3 p + 11 q$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "23 r plus 14 s plus 9 r plus open parenthesis −15 s close parenthesis .")[$23 r + 14 s + 9 r + ( −15 s ) .$]

#solutionbox[
32#emph[r] − #emph[s]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "37 m plus 21 n plus 4 m plus open parenthesis −15 n close parenthesis .")[$37 m + 21 n + 4 m + ( −15 n ) .$]

#solutionbox[
41#emph[m] + 6#emph[n]
]
]


=== Key Concepts

- #emph[Commutative Properties]- #emph[Commutative Property of Addition:]- If #math.equation(block: false, alt: "a , b")[$a , b$] are real numbers, then #math.equation(block: false, alt: "a plus b equals b plus a")[$a + b = b + a$]
- #emph[Commutative Property of Multiplication:]- If #math.equation(block: false, alt: "a , b")[$a , b$] are real numbers, then #math.equation(block: false, alt: "a times b equals b times a")[$a ⋅ b = b ⋅ a$]
- #emph[Associative Properties]- #emph[Associative Property of Addition:]- If #math.equation(block: false, alt: "a , b , c")[$a , b , c$] are real numbers then #math.equation(block: false, alt: "open parenthesis a plus b close parenthesis plus c equals a plus open parenthesis b plus c close parenthesis")[$( a + b ) + c = a + ( b + c )$]
- #emph[Associative Property of Multiplication:]- If #math.equation(block: false, alt: "a , b , c")[$a , b , c$] are real numbers then #math.equation(block: false, alt: "open parenthesis a times b close parenthesis times c equals a times open parenthesis b times c close parenthesis")[$( a ⋅ b ) ⋅ c = a ⋅ ( b ⋅ c )$]


==== Practice Makes Perfect

#strong[Use the Commutative and Associative Properties]

In the following exercises, use the commutative properties to rewrite the given expression.

#math.equation(block: true, alt: "8 plus 9 equals ___")[$8 + 9 = \_\_\_$]

#math.equation(block: true, alt: "7 plus 6 equals ___")[$7 + 6 = \_\_\_$]

#solutionbox[
7 + 6 = 6 + 7
]

#math.equation(block: true, alt: "8 open parenthesis −12 close parenthesis equals ___")[$8 ( −12 ) = \_\_\_$]

#math.equation(block: true, alt: "7 open parenthesis −13 close parenthesis equals ___")[$7 ( −13 ) = \_\_\_$]

#solutionbox[
7(−13) = (−13)7
]

#math.equation(block: true, alt: "open parenthesis −19 close parenthesis open parenthesis −14 close parenthesis equals ___")[$( −19 ) ( −14 ) = \_\_\_$]

#math.equation(block: true, alt: "open parenthesis −12 close parenthesis open parenthesis −18 close parenthesis equals ___")[$( −12 ) ( −18 ) = \_\_\_$]

#solutionbox[
(−12)(−18) = (−18)(−12)
]

#math.equation(block: true, alt: "−11 plus 8 equals ___")[$−11 + 8 = \_\_\_$]

#math.equation(block: true, alt: "−15 plus 7 equals ___")[$−15 + 7 = \_\_\_$]

#solutionbox[
−15 + 7 = 7 + (−15)
]

#math.equation(block: true, alt: "x plus 4 equals ___")[$x + 4 = \_\_\_$]

#math.equation(block: true, alt: "y plus 1 equals ___")[$y + 1 = \_\_\_$]

#solutionbox[
#emph[y] + 1 = 1 + #emph[y]
]

#math.equation(block: true, alt: "−2 a equals ___")[$−2 a = \_\_\_$]

#math.equation(block: true, alt: "−3 m equals ___")[$−3 m = \_\_\_$]

#solutionbox[
−3#emph[m] = #emph[m] (−3)
]

In the following exercises, use the associative properties to rewrite the given expression.

#math.equation(block: true, alt: "open parenthesis 11 plus 9 close parenthesis plus 14 equals ___")[$( 11 + 9 ) + 14 = \_\_\_$]

#math.equation(block: true, alt: "open parenthesis 21 plus 14 close parenthesis plus 9 equals ___")[$( 21 + 14 ) + 9 = \_\_\_$]

#solutionbox[
(21 + 14) + 9 = 21 + (14 + 9)
]

#math.equation(block: true, alt: "open parenthesis 12 times 5 close parenthesis times 7 equals ___")[$( 12 · 5 ) · 7 = \_\_\_$]

#math.equation(block: true, alt: "open parenthesis 14 times 6 close parenthesis times 9 equals ___")[$( 14 · 6 ) · 9 = \_\_\_$]

#solutionbox[
(14 · 6) · 9 = 14(6 · 9)
]

#math.equation(block: true, alt: "open parenthesis −7 plus 9 close parenthesis plus 8 equals ___")[$( −7 + 9 ) + 8 = \_\_\_$]

#math.equation(block: true, alt: "open parenthesis −2 plus 6 close parenthesis plus 7 equals ___")[$( −2 + 6 ) + 7 = \_\_\_$]

#solutionbox[
(−2 + 6) + 7 = −2 + (6 + 7)
]

#math.equation(block: true, alt: "open parenthesis 16 times the fraction 4 over 5 close parenthesis times 15 equals ___")[$( 16 · frac(4, 5) ) · 15 = \_\_\_$]

#math.equation(block: true, alt: "open parenthesis 13 times the fraction 2 over 3 close parenthesis times 18 equals ___")[$( 13 · frac(2, 3) ) · 18 = \_\_\_$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 13 times the fraction 2 over 3 close parenthesis times 18 equals 13 open parenthesis the fraction 2 over 3 times 18 close parenthesis")[$( 13 · frac(2, 3) ) · 18 = 13 ( frac(2, 3) · 18 )$]
]

#math.equation(block: true, alt: "3 open parenthesis 4 x close parenthesis equals ___")[$3 ( 4 x ) = \_\_\_$]

#math.equation(block: true, alt: "4 open parenthesis 7 x close parenthesis equals ___")[$4 ( 7 x ) = \_\_\_$]

#solutionbox[
4(7#emph[x]) = (4 · 7)#emph[x]
]

#math.equation(block: true, alt: "open parenthesis 12 plus x close parenthesis plus 28 equals ___")[$( 12 + x ) + 28 = \_\_\_$]

#math.equation(block: true, alt: "open parenthesis 17 plus y close parenthesis plus 33 equals ___")[$( 17 + y ) + 33 = \_\_\_$]

#solutionbox[
(17 + #emph[y]) + 33 = 17 + (#emph[y] + 33)
]

#strong[Evaluate Expressions using the Commutative and Associative Properties]

In the following exercises, evaluate each expression for the given value.

If #math.equation(block: false, alt: "y equals the fraction 5 over 8 ,")[$y = frac(5, 8) ,$] evaluate:

+ ⓐ #math.equation(block: false, alt: "y plus 0.49 plus open parenthesis minus y close parenthesis")[$#h(0.2em) y + 0.49 + ( − y )$]
+ ⓑ #math.equation(block: false, alt: "y plus open parenthesis minus y close parenthesis plus 0.49")[$#h(0.2em) y + ( − y ) + 0.49$]

If #math.equation(block: false, alt: "z equals the fraction 7 over 8 ,")[$z = frac(7, 8) ,$] evaluate:

- ⓐ #math.equation(block: false, alt: "z plus 0.97 plus open parenthesis minus z close parenthesis")[$#h(0.2em) z + 0.97 + ( − z )$]
- ⓑ #math.equation(block: false, alt: "z plus open parenthesis minus z close parenthesis plus 0.97")[$#h(0.2em) z + ( − z ) + 0.97$]

#solutionbox[
+ ⓐ 0.97
+ ⓑ 0.97
]

If #math.equation(block: false, alt: "c equals minus the fraction 11 over 4 ,")[$c = − frac(11, 4) ,$] evaluate:

+ ⓐ #math.equation(block: false, alt: "c plus 3.125 plus open parenthesis minus c close parenthesis")[$#h(0.2em) c + 3.125 + ( − c )$]
+ ⓑ #math.equation(block: false, alt: "c plus open parenthesis minus c close parenthesis plus 3.125")[$#h(0.2em) c + ( − c ) + 3.125$]

If #math.equation(block: false, alt: "d equals minus the fraction 9 over 4 ,")[$d = − frac(9, 4) ,$] evaluate:

+ ⓐ #math.equation(block: false, alt: "d plus 2.375 plus open parenthesis minus d close parenthesis")[$#h(0.2em) d + 2.375 + ( − d )$]
+ ⓑ #math.equation(block: false, alt: "d plus open parenthesis minus d close parenthesis plus 2.375")[$#h(0.2em) d + ( − d ) + 2.375$]

#solutionbox[
+ ⓐ 2.375
+ ⓑ 2.375
]

If #math.equation(block: false, alt: "j equals 11 ,")[$j = 11 ,$] evaluate:

+ ⓐ #math.equation(block: false, alt: "the fraction 5 over 6 open parenthesis the fraction 6 over 5 j close parenthesis")[$#h(0.2em) frac(5, 6) ( frac(6, 5) j )$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis the fraction 5 over 6 times the fraction 6 over 5 close parenthesis j")[$#h(0.2em) ( frac(5, 6) · frac(6, 5) ) j$]

If #math.equation(block: false, alt: "k equals 21 ,")[$k = 21 ,$] evaluate:

+ ⓐ #math.equation(block: false, alt: "the fraction 4 over 13 open parenthesis the fraction 13 over 4 k close parenthesis")[$#h(0.2em) frac(4, 13) ( frac(13, 4) k )$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis the fraction 4 over 13 times the fraction 13 over 4 close parenthesis k")[$#h(0.2em) ( frac(4, 13) · frac(13, 4) ) k$]

#solutionbox[
+ ⓐ 21
+ ⓑ 21
]

If #math.equation(block: false, alt: "m equals −25 ,")[$m = −25 ,$] evaluate:

+ ⓐ #math.equation(block: false, alt: "minus the fraction 3 over 7 open parenthesis the fraction 7 over 3 m close parenthesis")[$#h(0.2em) − frac(3, 7) ( frac(7, 3) m )$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis minus the fraction 3 over 7 times the fraction 7 over 3 close parenthesis m")[$#h(0.2em) ( − frac(3, 7) · frac(7, 3) ) m$]

If #math.equation(block: false, alt: "n equals −8 ,")[$n = −8 ,$] evaluate:

+ ⓐ #math.equation(block: false, alt: "minus the fraction 5 over 21 open parenthesis the fraction 21 over 5 n close parenthesis")[$#h(0.2em) − frac(5, 21) ( frac(21, 5) n )$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis minus the fraction 5 over 21 times the fraction 21 over 5 close parenthesis n")[$#h(0.2em) ( − frac(5, 21) · frac(21, 5) ) n$]

#solutionbox[
+ ⓐ 8
+ ⓑ 8
]

#strong[Simplify Expressions Using the Commutative and Associative Properties]

In the following exercises, simplify.

#math.equation(block: true, alt: "−45 a plus 15 plus 45 a")[$−45 a + 15 + 45 a$]

#math.equation(block: true, alt: "9 y plus 23 plus open parenthesis −9 y close parenthesis")[$9 y + 23 + ( −9 y )$]

#solutionbox[
23
]

#math.equation(block: true, alt: "the fraction 1 over 2 plus the fraction 7 over 8 plus open parenthesis minus the fraction 1 over 2 close parenthesis")[$frac(1, 2) + frac(7, 8) + ( − frac(1, 2) )$]

#math.equation(block: true, alt: "the fraction 2 over 5 plus the fraction 5 over 12 plus open parenthesis minus the fraction 2 over 5 close parenthesis")[$frac(2, 5) + frac(5, 12) + ( − frac(2, 5) )$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 over 12")[$frac(5, 12)$]
]

#math.equation(block: true, alt: "the fraction 3 over 20 times the fraction 49 over 11 times the fraction 20 over 3")[$frac(3, 20) · frac(49, 11) · frac(20, 3)$]

#math.equation(block: true, alt: "the fraction 13 over 18 times the fraction 25 over 7 times the fraction 18 over 13")[$frac(13, 18) · frac(25, 7) · frac(18, 13)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 25 over 7")[$frac(25, 7)$]
]

#math.equation(block: true, alt: "the fraction 7 over 12 times the fraction 9 over 17 times the fraction 24 over 7")[$frac(7, 12) · frac(9, 17) · frac(24, 7)$]

#math.equation(block: true, alt: "the fraction 3 over 10 times the fraction 13 over 23 times the fraction 50 over 3")[$frac(3, 10) · frac(13, 23) · frac(50, 3)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 65 over 23")[$frac(65, 23)$]
]

#math.equation(block: true, alt: "−24 times 7 times the fraction 3 over 8")[$−24 · 7 · frac(3, 8)$]

#math.equation(block: true, alt: "−36 times 11 times the fraction 4 over 9")[$−36 · 11 · frac(4, 9)$]

#solutionbox[
−176
]

#math.equation(block: true, alt: "open parenthesis the fraction 5 over 6 plus the fraction 8 over 15 close parenthesis plus the fraction 7 over 15")[$( frac(5, 6) + frac(8, 15) ) + frac(7, 15)$]

#math.equation(block: true, alt: "open parenthesis the fraction 1 over 12 plus the fraction 4 over 9 close parenthesis plus the fraction 5 over 9")[$( frac(1, 12) + frac(4, 9) ) + frac(5, 9)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 13 over 12")[$frac(13, 12)$]
]

#math.equation(block: true, alt: "the fraction 5 over 13 plus the fraction 3 over 4 plus the fraction 1 over 4")[$frac(5, 13) + frac(3, 4) + frac(1, 4)$]

#math.equation(block: true, alt: "the fraction 8 over 15 plus the fraction 5 over 7 plus the fraction 2 over 7")[$frac(8, 15) + frac(5, 7) + frac(2, 7)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 23 over 15")[$frac(23, 15)$]
]

#math.equation(block: true, alt: "open parenthesis 4.33 p plus 1.09 p close parenthesis plus 3.91 p")[$( 4.33 p + 1.09 p ) + 3.91 p$]

#math.equation(block: true, alt: "open parenthesis 5.89 d plus 2.75 d close parenthesis plus 1.25 d")[$( 5.89 d + 2.75 d ) + 1.25 d$]

#solutionbox[
9.89#emph[d]
]

#math.equation(block: true, alt: "17 open parenthesis 0.25 close parenthesis open parenthesis 4 close parenthesis")[$17 ( 0.25 ) ( 4 )$]

#math.equation(block: true, alt: "36 open parenthesis 0.2 close parenthesis open parenthesis 5 close parenthesis")[$36 ( 0.2 ) ( 5 )$]

#solutionbox[
36
]

#math.equation(block: true, alt: "[ 2.48 open parenthesis 12 close parenthesis ] open parenthesis 0.5 close parenthesis")[$[ 2.48 ( 12 ) ] ( 0.5 )$]

#math.equation(block: true, alt: "[ 9.731 open parenthesis 4 close parenthesis ] open parenthesis 0.75 close parenthesis")[$[ 9.731 ( 4 ) ] ( 0.75 )$]

#solutionbox[
29.193
]

#math.equation(block: true, alt: "7 open parenthesis 4 a close parenthesis")[$7 ( 4 a )$]

#math.equation(block: true, alt: "9 open parenthesis 8 w close parenthesis")[$9 ( 8 w )$]

#solutionbox[
72#emph[w]
]

#math.equation(block: true, alt: "−15 open parenthesis 5 m close parenthesis")[$−15 ( 5 m )$]

#math.equation(block: true, alt: "−23 open parenthesis 2 n close parenthesis")[$−23 ( 2 n )$]

#solutionbox[
−46#emph[n]
]

#math.equation(block: true, alt: "12 open parenthesis the fraction 5 over 6 p close parenthesis")[$12 ( frac(5, 6) p )$]

#math.equation(block: true, alt: "20 open parenthesis the fraction 3 over 5 q close parenthesis")[$20 ( frac(3, 5) q )$]

#solutionbox[
12#emph[q]
]

#math.equation(block: true, alt: "14 x plus 19 y plus 25 x plus 3 y")[$14 x + 19 y + 25 x + 3 y$]

#math.equation(block: true, alt: "15 u plus 11 v plus 27 u plus 19 v")[$15 u + 11 v + 27 u + 19 v$]

#solutionbox[
42#emph[u] + 30#emph[v]
]

#math.equation(block: true, alt: "43 m plus open parenthesis −12 n close parenthesis plus ​ open parenthesis −16 m close parenthesis plus open parenthesis −9 n close parenthesis")[$43 m + ( −12 n ) + "​" ( −16 m ) + ( −9 n )$]

#math.equation(block: true, alt: "−22 p plus 17 q plus open parenthesis −35 p close parenthesis plus open parenthesis −27 q close parenthesis")[$−22 p + 17 q + ( −35 p ) + ( −27 q )$]

#solutionbox[
−57#emph[p] + (−10#emph[q])
]

#math.equation(block: true, alt: "the fraction 3 over 8 g plus the fraction 1 over 12 h plus the fraction 7 over 8 g plus the fraction 5 over 12 h")[$frac(3, 8) g + frac(1, 12) h + frac(7, 8) g + frac(5, 12) h$]

#math.equation(block: true, alt: "the fraction 5 over 6 a plus the fraction 3 over 10 b plus the fraction 1 over 6 a plus the fraction 9 over 10 b")[$frac(5, 6) a + frac(3, 10) b + frac(1, 6) a + frac(9, 10) b$]

#solutionbox[
#math.equation(block: true, alt: "a plus the fraction 6 over 5 b")[$a + frac(6, 5) b$]
]

#math.equation(block: true, alt: "6.8 p plus 9.14 q plus open parenthesis −4.37 p close parenthesis plus open parenthesis −0.88 q close parenthesis")[$6.8 p + 9.14 q + ( −4.37 p ) + ( −0.88 q )$]

#math.equation(block: true, alt: "9.6 m plus 7.22 n plus open parenthesis −2.19 m close parenthesis plus open parenthesis −0.65 n close parenthesis")[$9.6 m + 7.22 n + ( −2.19 m ) + ( −0.65 n )$]

#solutionbox[
7.41#emph[m] + 6.57#emph[n]
]


==== Everyday Math

#strong[Stamps] Allie and Loren need to buy stamps. Allie needs four #math.equation(block: false, alt: "$0.49")[$"$0.49"$] stamps and nine #math.equation(block: false, alt: "$0.02")[$"$0.02"$] stamps. Loren needs eight #math.equation(block: false, alt: "$0.49")[$"$0.49"$] stamps and three #math.equation(block: false, alt: "$0.02")[$"$0.02"$] 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 #math.equation(block: false, alt: "$0.49")[$"$0.49"$] stamps do the girls need altogether? How much will they cost?
+ ⓔ How many #math.equation(block: false, alt: "$0.02")[$"$0.02"$] stamps do the girls need altogether? How much will they cost?

#strong[Counting Cash] Grant is totaling up the cash from a fundraising dinner. In one envelope, he has twenty-three #math.equation(block: false, alt: "$5")[$"$5"$] bills, eighteen #math.equation(block: false, alt: "$10")[$"$10"$] bills, and thirty-four #math.equation(block: false, alt: "$20")[$"$20"$] bills. In another envelope, he has fourteen #math.equation(block: false, alt: "$5")[$"$5"$] bills, nine #math.equation(block: false, alt: "$10")[$"$10"$] bills, and twenty-seven #math.equation(block: false, alt: "$20")[$"$20"$] 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 #math.equation(block: false, alt: "$5")[$"$5"$] bills?
+ ⓔ What is the value of all #math.equation(block: false, alt: "$10")[$"$10"$] bills?
+ ⓕ What is the value of all #math.equation(block: false, alt: "$20")[$"$20"$] bills?

#solutionbox[
+ ⓐ \$975
+ ⓑ \$700
+ ⓒ \$1675
+ ⓓ \$185
+ ⓔ \$270
+ ⓕ \$1220
]


==== Writing Exercises

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.

#solutionbox[
Answers will vary.
]


==== Self Check

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

#figure(figph[A self-assessment chart for students to rate their understanding of Commutative and Associative Properties, with options: Confidently, With some help, or No-I don't get it!], alt: "A self-assessment chart for students to rate their understanding of Commutative and Associative Properties, with options: Confidently, With some help, or No-I don't get it!", caption: none)

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