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== 3.4#h(0.6em)Multiply and Divide Integers

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

Translate the quotient of #math.equation(block: false, alt: "20")[$20$] and #math.equation(block: false, alt: "13")[$13$] into an algebraic expression. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "20 divided by 13")[$20 ÷ 13$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Add: #math.equation(block: false, alt: "−5 plus open parenthesis −5 close parenthesis plus open parenthesis −5 close parenthesis .")[$−5 + ( −5 ) + ( −5 ) .$] #linebreak() 
If you missed this problem, review .

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: false, alt: "Evaluate n plus 4 when n equals −7 .")[$"Evaluate" #h(0.2em) n + 4 #h(0.2em) "when" #h(0.2em) n = −7 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "minus 3")[$− 3$]
]
]


=== Multiply Integers

Since multiplication is mathematical shorthand for repeated addition, our counter model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction.

We remember that #math.equation(block: false, alt: "a times b")[$a · b$] means add #math.equation(block: false, alt: "a , b")[$a , #h(0.2em) b$] times. Here, we are using the model shown in  just to help us discover the pattern.

#figure(figph[This image has two columns. The first column has 5 times 3. Underneath, it states add 5, 3 times. Under this there are 3 rows of 5 blue circles labeled 15 positives and 5 times 3 equals 15. The second column has negative 5 times 3. Underneath it states add negative 5, 3 times. Under this there are 3 rows of 5 red circles labeled 15 negatives and negative 5 times 3 equals 15.], alt: "This image has two columns. The first column has 5 times 3. Underneath, it states add 5, 3 times. Under this there are 3 rows of 5 blue circles labeled 15 positives and 5 times 3 equals 15. The second column has negative 5 times 3. Underneath it states add negative 5, 3 times. Under this there are 3 rows of 5 red circles labeled 15 negatives and negative 5 times 3 equals 15.", caption: none)

Now consider what it means to multiply #math.equation(block: false, alt: "5")[$5$] by #math.equation(block: false, alt: "−3 .")[$−3 .$] It means subtract #math.equation(block: false, alt: "5 , 3")[$5 , 3$] times. Looking at subtraction as #emph[taking away], it means to take away #math.equation(block: false, alt: "5 , 3")[$5 , 3$] times. But there is nothing to take away, so we start by adding neutral pairs as shown.

#figure(figph[This figure has 2 columns. The first column has 5 times negative 3. Underneath it states take away 5, 3 times. Under this there are 3 rows of 5 red circles. A downward arrow points to six rows of alternating colored circles in rows of fives. The first row includes 5 red circles, followed by five blue circles, then 5 red, five blue, five red, and five blue. All of the rows of blue circles are circled. The non-circled rows are labeled 15 negatives.  Under the label is 5 times negative 3 equals negative 15. The second column has negative 5 times negative 3. Underneath it states take away negative 5, 3 times. Then there are 6 rows of 5 circles alternating in color. The first row is 5 blue circles followed by 5 red circles. All of the red rows are circled. The non-circles rows are labeled 15 positives. Under the label is negative 5 times negative 3 equals 15.], alt: "This figure has 2 columns. The first column has 5 times negative 3. Underneath it states take away 5, 3 times. Under this there are 3 rows of 5 red circles. A downward arrow points to six rows of alternating colored circles in rows of fives. The first row includes 5 red circles, followed by five blue circles, then 5 red, five blue, five red, and five blue. All of the rows of blue circles are circled. The non-circled rows are labeled 15 negatives. Under the label is 5 times negative 3 equals negative 15. The second column has negative 5 times negative 3. Underneath it states take away negative 5, 3 times. Then there are 6 rows of 5 circles alternating in color. The first row is 5 blue circles followed by 5 red circles. All of the red rows are circled. The non-circles rows are labeled 15 positives. Under the label is negative 5 times negative 3 equals 15.", caption: none)

In both cases, we started with #math.equation(block: false, alt: "15")[$"15"$] neutral pairs. In the case on the left, we took away #math.equation(block: false, alt: "5 , 3")[$"5" , "3"$] times and the result was #math.equation(block: false, alt: "minus 15 .")[$− "15" .$] To multiply #math.equation(block: false, alt: "open parenthesis −5 close parenthesis open parenthesis −3 close parenthesis ,")[$( −5 ) ( −3 ) ,$] we took away #math.equation(block: false, alt: "minus 5 , 3")[$− "5" , "3"$] times and the result was #math.equation(block: false, alt: "15 .")[$"15" .$] So we found that

#math.equation(block: true, alt: "5 times 3 equals 15, −5 open parenthesis 3 close parenthesis equals −15; 5 open parenthesis −3 close parenthesis equals −15, open parenthesis −5 close parenthesis open parenthesis −3 close parenthesis equals 15")[$5 · 3 = 15 &  & −5 ( 3 ) = −15 \ 5 ( −3 ) = −15 &  & ( −5 ) ( −3 ) = 15$]Notice that for multiplication of two #strong[signed numbers], when the signs are the same, the #strong[product] is positive, and when the signs are different, the product is negative.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Multiplication of Signed Numbers]

The sign of the product of two numbers depends on their signs.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Same signs], [Product]),
  [⋅Two positives #linebreak() ⋅Two negatives], [Positive #linebreak() Positive],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Different signs], [Product]),
  [⋅Positive ⋅ negative #linebreak() ⋅Negative ⋅ positive], [Negative #linebreak() Negative],
))
]

#examplebox("Example 1")[][
Multiply each of the following: #linebreak()

+ ⓐ #math.equation(block: false, alt: "−9 times 3")[$#h(0.2em) −9 · 3 #h(1em)$]
+ ⓑ #math.equation(block: false, alt: "−2 open parenthesis −5 close parenthesis")[$#h(0.2em) −2 ( −5 ) #h(1em)$]
+ ⓒ #math.equation(block: false, alt: "4 open parenthesis −8 close parenthesis")[$#h(0.2em) 4 ( −8 ) #h(1em)$]
+ ⓓ #math.equation(block: false, alt: "7 times 6")[$#h(0.2em) 7 · 6$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "–9 times 3")[$–9 ⋅ 3$]],
  [Multiply, noting that the signs are different and so the product is negative.], [#math.equation(block: false, alt: "–27")[$–27$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "–2 open parenthesis –5 close parenthesis")[$–2 ( –5 )$]],
  [Multiply, noting that the signs are the same and so the product is positive.], [#math.equation(block: false, alt: "10")[$10$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓒ], []),
  [], [#math.equation(block: false, alt: "4 open parenthesis –8 close parenthesis")[$4 ( –8 )$]],
  [Multiply, noting that the signs are different and so the product is negative.], [#math.equation(block: false, alt: "–32")[$–32$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓓ], []),
  [], [#math.equation(block: false, alt: "7 times 6")[$7 ⋅ 6$]],
  [The signs are the same, so the product is positive.], [#math.equation(block: false, alt: "42")[$42$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply:

+ ⓐ #math.equation(block: false, alt: "−6 times 8")[$#h(0.2em) −6 · 8 #h(1em)$]
+ ⓑ #math.equation(block: false, alt: "−4 open parenthesis −7 close parenthesis")[$#h(0.2em) −4 ( −7 ) #h(1em)$]
+ ⓒ #math.equation(block: false, alt: "9 open parenthesis −7 close parenthesis")[$#h(0.2em) 9 ( −7 ) #h(1em)$]
+ ⓓ #math.equation(block: false, alt: "5 times 12")[$#h(0.2em) 5 · 12$]

#solutionbox[
+ ⓐ −48
+ ⓑ 28
+ ⓒ −63
+ ⓓ 60
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply:

+ ⓐ #math.equation(block: false, alt: "−8 times 7")[$#h(0.2em) −8 · 7 #h(1em)$]
+ ⓑ #math.equation(block: false, alt: "−6 open parenthesis −9 close parenthesis")[$#h(0.2em) −6 ( −9 ) #h(1em)$]
+ ⓒ #math.equation(block: false, alt: "7 open parenthesis −4 close parenthesis")[$#h(0.2em) 7 ( −4 ) #h(1em)$]
+ ⓓ #math.equation(block: false, alt: "3 times 13")[$#h(0.2em) 3 · 13$]

#solutionbox[
+ ⓐ −56
+ ⓑ 54
+ ⓒ −28
+ ⓓ 39
]
]

When we multiply a number by #math.equation(block: false, alt: "1 ,")[$1 ,$] the result is the same number. What happens when we multiply a number by#math.equation(block: false, alt: "−1 ?")[$−1 ?$] Let’s multiply a positive number and then a negative number by #math.equation(block: false, alt: "−1")[$−1$] to see what we get.

#math.equation(block: true, alt: "−1 times 4, , −1 open parenthesis −3 close parenthesis; −4, , 3; −4 is the opposite of 4, , 3 is the opposite of −3")[$−1 · 4 & #h(2em) & −1 ( −3 ) \ −4 & #h(2em) & 3 \ −4 #h(0.2em) "is the opposite of" #h(0.2em) "4" & #h(2em) & "3" #h(0.2em) "is the opposite of" #h(0.2em) −3$]Each time we multiply a number by #math.equation(block: false, alt: "−1 ,")[$−1 ,$] we get its opposite.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Multiplication by #math.equation(block: false, alt: "−1")[$−1$]]

Multiplying a number by #math.equation(block: false, alt: "−1")[$−1$] gives its opposite.

#math.equation(block: true, alt: "−1 a equals minus a")[$−1 a = − a$]
]

#examplebox("Example 2")[][
Multiply each of the following:

+ ⓐ #math.equation(block: false, alt: "−1 times 7")[$#h(0.2em) −1 · 7 #h(1em)$]
+ ⓑ #math.equation(block: false, alt: "−1 open parenthesis −11 close parenthesis")[$#h(0.2em) −1 ( −11 )$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [The signs are different, so the product will be negative.], [#math.equation(block: false, alt: "−1 times 7")[$−1 ⋅ 7$]],
  [Notice that −7 is the opposite of 7.], [#math.equation(block: false, alt: "−7")[$−7$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [The signs are the same, so the product will be positive.], [#math.equation(block: false, alt: "−1 open parenthesis −11 close parenthesis")[$−1 ( −11 )$]],
  [Notice that 11 is the opposite of −11.], [#math.equation(block: false, alt: "11")[$11$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply.

+ ⓐ #math.equation(block: false, alt: "−1 times 9")[$#h(0.2em) −1 · 9 #h(1em)$]
+ ⓑ #math.equation(block: false, alt: "−1 times open parenthesis −17 close parenthesis")[$#h(0.2em) −1 · ( −17 )$]

#solutionbox[
+ ⓐ −9
+ ⓑ 17
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply.

+ ⓐ #math.equation(block: false, alt: "−1 times 8")[$#h(0.2em) −1 · 8 #h(1em)$]
+ ⓑ #math.equation(block: false, alt: "−1 times open parenthesis −16 close parenthesis")[$#h(0.2em) −1 · ( −16 )$]

#solutionbox[
+ ⓐ −8
+ ⓑ 16
]
]


=== Divide Integers

Division is the inverse operation of multiplication. So, #math.equation(block: false, alt: "15 divided by 3 equals 5")[$15 ÷ 3 = 5$] because #math.equation(block: false, alt: "5 times 3 equals 15")[$5 · 3 = 15$] In words, this expression says that #math.equation(block: false, alt: "15")[$"15"$] can be divided into #math.equation(block: false, alt: "3")[$"3"$] groups of #math.equation(block: false, alt: "5")[$"5"$] each because adding five three times gives #math.equation(block: false, alt: "15 .")[$"15" .$] If we look at some examples of #strong[multiplying integers], we might figure out the rules for #strong[dividing integers].

#math.equation(block: true, alt: "5 times 3 equals 15 so 15 divided by 3 equals 5, −5 open parenthesis 3 close parenthesis equals −15 so −15 divided by 3 equals −5; open parenthesis −5 close parenthesis open parenthesis −3 close parenthesis equals 15 so 15 divided by open parenthesis −3 close parenthesis equals −5, 5 open parenthesis −3 close parenthesis equals −15 so −15 divided by −3 equals 5")[$5 · 3 = 15 #h(0.2em) "so" #h(0.2em) 15 ÷ 3 = 5 &  &  &  & −5 ( 3 ) = −15 #h(0.2em) "so" #h(0.2em) −15 ÷ 3 = −5 \ ( −5 ) ( −3 ) = 15 #h(0.2em) "so" #h(0.2em) 15 ÷ ( −3 ) = −5 &  &  &  & 5 ( −3 ) = −15 #h(0.2em) "so" #h(0.2em) −15 ÷ −3 = 5$]Division of signed numbers follows the same rules as multiplication. When the signs are the same, the #strong[quotient] is positive, and when the signs are different, the quotient is negative.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Division of Signed Numbers]

The sign of the quotient of two numbers depends on their signs.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Same signs], [Quotient]),
  [⋅Two positives #linebreak() ⋅Two negatives], [Positive #linebreak() Positive],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Different signs], [Quotient]),
  [⋅Positive & negative #linebreak() ⋅Negative & positive], [Negative #linebreak() Negative],
))
]

Remember, you can always check the answer to a division problem by multiplying.

#examplebox("Example 3")[][
Divide each of the following: #linebreak()

+ ⓐ #math.equation(block: false, alt: "−27 divided by 3")[$#h(0.2em) −27 ÷ 3 #h(1em)$]
+ ⓑ #math.equation(block: false, alt: "−100 divided by open parenthesis −4 close parenthesis")[$#h(0.2em) −100 ÷ ( −4 )$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "–27 divided by 3")[$–27 ÷ 3$]],
  [Divide, noting that the signs are different and so the quotient is negative.], [#math.equation(block: false, alt: "–9")[$–9$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "–100 divided by open parenthesis –4 close parenthesis")[$–100 ÷ ( –4 )$]],
  [Divide, noting that the signs are the same and so the quotient is positive.], [#math.equation(block: false, alt: "25")[$25$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide:

+ ⓐ #math.equation(block: false, alt: "−42 divided by 6")[$#h(0.2em) −42 ÷ 6 #h(1em)$]
+ ⓑ #math.equation(block: false, alt: "−117 divided by open parenthesis −3 close parenthesis")[$#h(0.2em) −117 ÷ ( −3 )$]

#solutionbox[
+ ⓐ −7
+ ⓑ 39
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide:

+ ⓐ #math.equation(block: false, alt: "−63 divided by 7")[$#h(0.2em) −63 ÷ 7 #h(1em)$]
+ ⓑ #math.equation(block: false, alt: "−115 divided by open parenthesis −5 close parenthesis")[$#h(0.2em) −115 ÷ ( −5 )$]

#solutionbox[
+ ⓐ −9
+ ⓑ 23
]
]

Just as we saw with multiplication, when we divide a number by #math.equation(block: false, alt: "1 ,")[$1 ,$] the result is the same number. What happens when we divide a number by #math.equation(block: false, alt: "−1 ?")[$−1 ?$] Let’s divide a positive number and then a negative number by #math.equation(block: false, alt: "−1")[$−1$] to see what we get.

#math.equation(block: true, alt: "8 divided by open parenthesis −1 close parenthesis, −9 divided by open parenthesis −1 close parenthesis; −8, 9; −8 is the opposite of 8, 9 is the opposite of −9")[$8 ÷ ( −1 ) &  &  & −9 ÷ ( −1 ) \ −8 &  &  & 9 \ "−8 is the opposite of 8" &  &  & "9 is the opposite of −9"$]When we divide a number by, #math.equation(block: false, alt: "−1")[$−1$] we get its opposite.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Division by #math.equation(block: false, alt: "−1")[$−1$]]

Dividing a number by #math.equation(block: false, alt: "−1")[$−1$] gives its opposite.

#math.equation(block: true, alt: "a divided by open parenthesis −1 close parenthesis equals −a")[$a ÷ ( −1 ) = "−a"$]
]

#examplebox("Example 4")[][
Divide each of the following: #linebreak()

+ ⓐ #math.equation(block: false, alt: "16 divided by open parenthesis −1 close parenthesis")[$#h(0.2em) 16 ÷ ( −1 ) #h(1em)$]
+ ⓑ #math.equation(block: false, alt: "−20 divided by open parenthesis −1 close parenthesis")[$#h(0.2em) −20 ÷ ( −1 )$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#math.equation(block: false, alt: "16 divided by open parenthesis –1 close parenthesis")[$16 ÷ ( –1 )$]],
  [The dividend, 16, is being divided by –1.], [#math.equation(block: false, alt: "–16")[$–16$]],
  [Dividing a number by –1 gives its opposite.], [],
  [Notice that the signs were different, so the result was negative.], [],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#math.equation(block: false, alt: "–20 divided by open parenthesis –1 close parenthesis")[$–20 ÷ ( –1 )$]],
  [The dividend, –20, is being divided by –1.], [#math.equation(block: false, alt: "20")[$20$]],
  [Dividing a number by –1 gives its opposite.], [],
))

Notice that the signs were the same, so the quotient was positive.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide:

+ ⓐ 
#math.equation(block: false, alt: "6 divided by open parenthesis −1 close parenthesis")[$#h(0.2em) 6 ÷ ( −1 ) #h(1em)$]
+ ⓑ 
#math.equation(block: false, alt: "−36 divided by open parenthesis −1 close parenthesis")[$#h(0.2em) −36 ÷ ( −1 )$]

#solutionbox[
+ ⓐ −6
+ ⓑ 36
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Divide:

+ ⓐ 
#math.equation(block: false, alt: "28 divided by open parenthesis −1 close parenthesis")[$#h(0.2em) 28 ÷ ( −1 ) #h(1em)$]
+ ⓑ 
#math.equation(block: false, alt: "−52 divided by open parenthesis −1 close parenthesis")[$#h(0.2em) −52 ÷ ( −1 )$]

#solutionbox[
+ ⓐ −28
+ ⓑ 52
]
]


=== Simplify Expressions with Integers

Now we’ll simplify #strong[expressions] that use all four operations–addition, subtraction, multiplication, and division–with integers. Remember to follow the order of operations.

#examplebox("Example 5")[][
#math.equation(block: true, alt: "Simplify: 7 open parenthesis −2 close parenthesis plus 4 open parenthesis −7 close parenthesis minus 6 .")[$"Simplify:" #h(0.2em) 7 ( −2 ) + 4 ( −7 ) − 6 .$]

#solutionbox[
We use the order of operations. Multiply first and then add and subtract from left to right.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "7 open parenthesis −2 close parenthesis plus 4 open parenthesis −7 close parenthesis −6")[$7 ( −2 ) + 4 ( −7 ) −6$]]),
  [Multiply first.], [#math.equation(block: false, alt: "−14 plus open parenthesis −28 close parenthesis −6")[$−14 + ( −28 ) −6$]],
  [Add.], [#math.equation(block: false, alt: "−42 minus 6")[$−42 − 6$]],
  [Subtract.], [#math.equation(block: false, alt: "−48")[$−48$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:

#math.equation(block: true, alt: "8 open parenthesis −3 close parenthesis plus 5 open parenthesis −7 close parenthesis −4")[$8 ( −3 ) + 5 ( −7 ) −4$]

#solutionbox[
−63
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:

#math.equation(block: true, alt: "9 open parenthesis −3 close parenthesis plus 7 open parenthesis −8 close parenthesis minus 1")[$9 ( −3 ) + 7 ( −8 ) − 1$]

#solutionbox[
−84
]
]

#examplebox("Example 6")[][
Simplify:

+ ⓐ 
#math.equation(block: false, alt: "open parenthesis −2 close parenthesis to the power 4")[$#h(0.2em) attach(( −2 ), t: 4) #h(1em)$]
+ ⓑ 
#math.equation(block: false, alt: "−2 to the power 4")[$#h(0.2em) −2^(4)$]

#solutionbox[
The exponent tells how many times to multiply the base.

ⓐ The exponent is #math.equation(block: false, alt: "4")[$4$] and the base is #math.equation(block: false, alt: "−2 .")[$−2 .$] We raise #math.equation(block: false, alt: "−2")[$−2$] to the fourth power.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis −2 close parenthesis to the power 4")[$attach(( −2 ), t: 4)$]]),
  [Write in expanded form.], [#math.equation(block: false, alt: "open parenthesis −2 close parenthesis open parenthesis −2 close parenthesis open parenthesis −2 close parenthesis open parenthesis −2 close parenthesis")[$( −2 ) ( −2 ) ( −2 ) ( −2 )$]],
  [Multiply.], [#math.equation(block: false, alt: "4 open parenthesis −2 close parenthesis open parenthesis −2 close parenthesis")[$4 ( −2 ) ( −2 )$]],
  [Multiply.], [#math.equation(block: false, alt: "minus 8 open parenthesis −2 close parenthesis")[$− 8 ( −2 )$]],
  [Multiply.], [#math.equation(block: false, alt: "16")[$16$]],
))

ⓑ The exponent is #math.equation(block: false, alt: "4")[$4$] and the base is #math.equation(block: false, alt: "2 .")[$2 .$] We raise #math.equation(block: false, alt: "2")[$2$] to the fourth power and then take the opposite.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "minus 2 to the power 4")[$− 2^(4)$]]),
  [Write in expanded form.], [#math.equation(block: false, alt: "minus open parenthesis 2 times 2 times 2 times 2 close parenthesis")[$− ( 2 ⋅ 2 ⋅ 2 ⋅ 2 )$]],
  [Multiply.], [#math.equation(block: false, alt: "minus open parenthesis 4 times 2 times 2 close parenthesis")[$− ( 4 ⋅ 2 ⋅ 2 )$]],
  [Multiply.], [#math.equation(block: false, alt: "minus open parenthesis 8 times 2 close parenthesis")[$− ( 8 ⋅ 2 )$]],
  [Multiply.], [#math.equation(block: false, alt: "minus 16")[$− 16$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:

+ ⓐ 
#math.equation(block: false, alt: "open parenthesis −3 close parenthesis to the power 4")[$#h(0.2em) attach(( −3 ), t: 4) #h(1em)$]
+ ⓑ 
#math.equation(block: false, alt: "−3 to the power 4")[$#h(0.2em) −3^(4)$]

#solutionbox[
+ ⓐ 81
+ ⓑ −81
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:

+ ⓐ 
#math.equation(block: false, alt: "open parenthesis −7 close parenthesis squared")[$#h(0.2em) attach(( −7 ), t: 2) #h(1em)$]
+ ⓑ 
#math.equation(block: false, alt: "minus 7 squared")[$#h(0.2em) − 7^(2)$]

#solutionbox[
+ ⓐ 49
+ ⓑ −49
]
]

#examplebox("Example 7")[][
#math.equation(block: true, alt: "Simplify: 12 minus 3 open parenthesis 9 minus 12 close parenthesis .")[$"Simplify:" #h(0.2em) 12 − 3 ( 9 − 12 ) .$]

#solutionbox[
According to the order of operations, we simplify inside parentheses first. Then we will multiply and finally we will subtract.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "12 minus 3 open parenthesis 9 minus 12 close parenthesis")[$12 − 3 ( 9 − 12 )$]]),
  [Subtract the parentheses first.], [#math.equation(block: false, alt: "12 minus 3 open parenthesis −3 close parenthesis")[$12 − 3 ( −3 )$]],
  [Multiply.], [#math.equation(block: false, alt: "12 minus open parenthesis −9 close parenthesis")[$12 − ( −9 )$]],
  [Subtract.], [#math.equation(block: false, alt: "21")[$21$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:

#math.equation(block: true, alt: "17 minus 4 open parenthesis 8 minus 11 close parenthesis")[$17 − 4 ( 8 − 11 )$]

#solutionbox[
29
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:

#math.equation(block: true, alt: "16 minus 6 open parenthesis 7 minus 13 close parenthesis")[$16 − 6 ( 7 − 13 )$]

#solutionbox[
52
]
]

#examplebox("Example 8")[][
Simplify: #math.equation(block: false, alt: "8 open parenthesis −9 close parenthesis divided by open parenthesis −2 close parenthesis cubed .")[$8 ( −9 ) ÷ attach(( −2 ), t: 3) .$]

#solutionbox[
We simplify the exponent first, then multiply and divide.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "8 open parenthesis −9 close parenthesis divided by open parenthesis −2 close parenthesis cubed")[$8 ( −9 ) ÷ attach(( −2 ), t: 3)$]]),
  [Simplify the exponent.], [#math.equation(block: false, alt: "8 open parenthesis −9 close parenthesis divided by open parenthesis −8 close parenthesis")[$8 ( −9 ) ÷ ( −8 )$]],
  [Multiply.], [#math.equation(block: false, alt: "−72 divided by open parenthesis −8 close parenthesis")[$−72 ÷ ( −8 )$]],
  [Divide.], [#math.equation(block: false, alt: "9")[$9$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:

#math.equation(block: true, alt: "12 open parenthesis −9 close parenthesis divided by open parenthesis −3 close parenthesis cubed")[$12 ( −9 ) ÷ attach(( −3 ), t: 3)$]

#solutionbox[
4
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:

#math.equation(block: true, alt: "18 open parenthesis −4 close parenthesis divided by open parenthesis −2 close parenthesis cubed")[$18 ( −4 ) ÷ attach(( −2 ), t: 3)$]

#solutionbox[
9
]
]

#examplebox("Example 9")[][
#math.equation(block: true, alt: "Simplify: −30 divided by 2 plus open parenthesis −3 close parenthesis open parenthesis −7 close parenthesis .")[$"Simplify:" #h(0.2em) −30 ÷ 2 + ( −3 ) ( −7 ) .$]

#solutionbox[
First we will multiply and divide from left to right. Then we will add.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "−30 divided by 2 plus open parenthesis −3 close parenthesis open parenthesis −7 close parenthesis")[$−30 ÷ 2 + ( −3 ) ( −7 )$]]),
  [Divide.], [#math.equation(block: false, alt: "−15 plus open parenthesis −3 close parenthesis open parenthesis −7 close parenthesis")[$−15 + ( −3 ) ( −7 )$]],
  [Multiply.], [#math.equation(block: false, alt: "−15 plus 21")[$−15 + 21$]],
  [Add.], [#math.equation(block: false, alt: "6")[$6$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:

#math.equation(block: true, alt: "−27 divided by 3 plus open parenthesis −5 close parenthesis open parenthesis −6 close parenthesis")[$−27 ÷ 3 + ( −5 ) ( −6 )$]

#solutionbox[
21
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify:

#math.equation(block: true, alt: "−32 divided by 4 plus open parenthesis −2 close parenthesis open parenthesis −7 close parenthesis")[$−32 ÷ 4 + ( −2 ) ( −7 )$]

#solutionbox[
6
]
]


=== Evaluate Variable Expressions with Integers

Now we can evaluate #strong[expressions] that include multiplication and division with integers. Remember that to evaluate an expression, substitute the numbers in place of the variables, and then simplify.

#examplebox("Example 10")[][
#math.equation(block: true, alt: "Evaluate 2 x squared minus 3 x plus 8 when x equals −4 .")[$"Evaluate" #h(0.2em) 2 x^(2) − 3 x + 8 #h(0.2em) "when" #h(0.2em) x = −4 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression, 2x^2 - 3x + 8, is shown against a white background.], alt: "A mathematical expression, 2x^2 - 3x + 8, is shown against a white background.", caption: none)]),
  [#figure(figph[The text 'Substitute -4 for x.' is displayed, with '-4' highlighted in red, indicating a numerical substitution instruction.], alt: "The text 'Substitute -4 for x.' is displayed, with '-4' highlighted in red, indicating a numerical substitution instruction.", caption: none)], [#figure(figph[A mathematical expression: 2(-4)^2 - 3(-4) + 8, with the number -4 highlighted in red within parentheses.], alt: "A mathematical expression: 2(-4)^2 - 3(-4) + 8, with the number -4 highlighted in red within parentheses.", caption: none)],
  [Simplify exponents.], [#figure(figph[A mathematical expression showing the calculation 2(16) - 3(-4) + 8.], alt: "A mathematical expression showing the calculation 2(16) - 3(-4) + 8.", caption: none)],
  [Multiply.], [#figure(figph[A mathematical expression showing '32 minus negative 12 plus 8' on a white background. This problem involves integer operations, specifically subtraction of a negative number which becomes addition, followed by addition.], alt: "A mathematical expression showing '32 minus negative 12 plus 8' on a white background. This problem involves integer operations, specifically subtraction of a negative number which becomes addition, followed by addition.", caption: none)],
  [Subtract.], [#figure(figph[The mathematical expression '44 + 8' is displayed in black text against a plain white background.], alt: "The mathematical expression '44 + 8' is displayed in black text against a plain white background.", caption: none)],
  [Add.], [#figure(figph[The number '52' is displayed in black text on a plain white background, positioned towards the right side of the image.], alt: "The number '52' is displayed in black text on a plain white background, positioned towards the right side of the image.", caption: none)],
))

Keep in mind that when we substitute #math.equation(block: false, alt: "−4")[$−4$] for #math.equation(block: false, alt: "x ,")[$x ,$] we use parentheses to show the multiplication. Without parentheses, it would look like #math.equation(block: false, alt: "2 times −4 squared minus 3 times −4 plus 8 .")[$2 · −4^(2) − 3 · −4 + 8 .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate:

#math.equation(block: true, alt: "3 x squared minus 2 x plus 6 when x equals −3")[$3 x^(2) − 2 x + 6 #h(0.2em) "when" #h(0.2em) x = −3$]

#solutionbox[
39
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate:

#math.equation(block: true, alt: "4 x squared minus x minus 5 when x equals −2")[$4 x^(2) − x − 5 #h(0.2em) "when" #h(0.2em) x = −2$]

#solutionbox[
13
]
]

#examplebox("Example 11")[][
#math.equation(block: true, alt: "Evaluate 3 x plus 4 y minus 6 when x equals −1 and y equals 2 .")[$"Evaluate" #h(0.2em) 3 x + 4 y − 6 #h(0.2em) "when" #h(0.2em) x = −1 #h(0.2em) "and" #h(0.2em) y = 2 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The mathematical expression '3x + 4y - 6' is displayed in black text on a white background.], alt: "The mathematical expression '3x + 4y - 6' is displayed in black text on a white background.", caption: none)]),
  [Substitute #math.equation(block: false, alt: "x equals −1")[$x = −1$] and #math.equation(block: false, alt: "y equals 2")[$y = 2$].], [#figure(figph[A mathematical expression reads 3 multiplied by negative 1, plus 4 multiplied by 2, minus 6. The number -1 is highlighted in red, and 2 is highlighted in blue.], alt: "A mathematical expression reads 3 multiplied by negative 1, plus 4 multiplied by 2, minus 6. The number -1 is highlighted in red, and 2 is highlighted in blue.", caption: none)],
  [Multiply.], [#figure(figph[A mathematical expression showing the addition and subtraction of integers: -3 + 8 - 6.], alt: "A mathematical expression showing the addition and subtraction of integers: -3 + 8 - 6.", caption: none)],
  [Simplify.], [#figure(figph[The mathematical expression '-1' is displayed in a dark gray font against a plain white background.], alt: "The mathematical expression '-1' is displayed in a dark gray font against a plain white background.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate:

#math.equation(block: true, alt: "7 x plus 6 y minus 12 when x equals −2 and y equals 3")[$7 x + 6 y − 12 #h(0.2em) "when" #h(0.2em) x = −2 #h(0.2em) "and" #h(0.2em) y = 3$]

#solutionbox[
−8
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate:

#math.equation(block: true, alt: "8 x minus 6 y plus 13 when x equals −3 and y equals −5")[$8 x − 6 y + 13 #h(0.2em) "when" #h(0.2em) x = −3 #h(0.2em) "and" #h(0.2em) y = −5$]

#solutionbox[
19
]
]


=== Translate Word Phrases to Algebraic Expressions

Once again, all our prior work translating words to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is #emph[product] and for division is #emph[quotient].

#examplebox("Example 12")[][
Translate to an algebraic expression and simplify if possible: the product of #math.equation(block: false, alt: "−2")[$−2$] and #math.equation(block: false, alt: "14 .")[$14 .$]

#solutionbox[
The word #emph[product] tells us to multiply.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [the product of #math.equation(block: false, alt: "−2")[$−2$] and #math.equation(block: false, alt: "14")[$14$]]),
  [Translate.], [#math.equation(block: false, alt: "open parenthesis −2 close parenthesis open parenthesis 14 close parenthesis")[$( −2 ) ( 14 )$]],
  [Simplify.], [#math.equation(block: false, alt: "−28")[$−28$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate to an algebraic expression and simplify if possible:

#math.equation(block: true, alt: "the product of −5 and 12")[$"the product of −5 and 12"$]

#solutionbox[
−5 (12) = −60
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate to an algebraic expression and simplify if possible:

#math.equation(block: true, alt: "the product of 8 and −13")[$"the product of 8 and −13"$]

#solutionbox[
8 (−13) = −104
]
]

#examplebox("Example 13")[][
Translate to an algebraic expression and simplify if possible: the quotient of #math.equation(block: false, alt: "−56")[$−56$] and #math.equation(block: false, alt: "−7 .")[$−7 .$]

#solutionbox[
The word #emph[quotient] tells us to divide.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [the quotient of −56 and −7]),
  [Translate.], [#math.equation(block: false, alt: "−56 divided by open parenthesis −7 close parenthesis")[$−56 ÷ ( −7 )$]],
  [Simplify.], [#math.equation(block: false, alt: "8")[$8$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate to an algebraic expression and simplify if possible:

#math.equation(block: true, alt: "the quotient of −63 and −9")[$"the quotient of −63 and −9"$]

#solutionbox[
−63 ÷ −9 = 7
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate to an algebraic expression and simplify if possible:

#math.equation(block: true, alt: "the quotient of −72 and −9")[$"the quotient of −72 and −9"$]

#solutionbox[
−72 ÷ −9 = 8
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[ACCESS ADDITIONAL ONLINE RESOURCES]

- #link("https://openstaxcollege.org/l/24Multiplyinteg")[Multiplying Integers Using Color Counters]
- #link("https://openstaxcollege.org/l/24Multiplyneutr")[Multiplying Integers Using Color Counters With Neutral Pairs]
- #link("https://openstaxcollege.org/l/24Multiplybasic")[Multiplying Integers Basics]
- #link("https://openstaxcollege.org/l/24Dividebasic")[Dividing Integers Basics]
- #link("https://openstaxcollege.org/l/24Divideinteger")[Ex. Dividing Integers]
- #link("https://openstaxcollege.org/l/24Multidivisign")[Multiplying and Dividing Signed Numbers]
]


=== Key Concepts

- #strong[Multiplication of Signed Numbers]

- To determine the sign of the product of two signed numbers:#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Same Signs], [Product]),
  [Two positives #linebreak() Two negatives], [Positive #linebreak() Positive],
))

 #linebreak() #figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Different Signs], [Product]),
  [Positive ⋅ negative #linebreak() Negative ⋅ positive], [Negative #linebreak() Negative],
))
- #strong[Division of Signed Numbers]

- To determine the sign of the quotient of two signed numbers:#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Same Signs], [Quotient]),
  [Two positives #linebreak() Two negatives], [Positive #linebreak() Positive],
))

 #linebreak() #figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Different Signs], [Quotient]),
  [Positive ⋅ negative #linebreak() Negative ⋅ Positive], [Negative #linebreak() Negative],
))
- #strong[Multiplication by #math.equation(block: false, alt: "−1")[$−1$]]
- Multiplying a number by #math.equation(block: false, alt: "−1")[$−1$] gives its opposite: #math.equation(block: false, alt: "−1 a equals minus a")[$−1 a = − a$]
- #strong[Division by #math.equation(block: false, alt: "−1")[$−1$]]
- Dividing a number by #math.equation(block: false, alt: "−1")[$−1$] gives its opposite: #math.equation(block: false, alt: "a divided by open parenthesis −1 close parenthesis equals −a")[$a ÷ ( −1 ) = "−a"$]


==== Practice Makes Perfect

#strong[Multiply Integers]

In the following exercises, multiply each pair of integers.

#math.equation(block: true, alt: "−4 times 8")[$−4 · 8$]

#solutionbox[
−32
]

#math.equation(block: true, alt: "−3 times 9")[$−3 · 9$]

#math.equation(block: true, alt: "−5 open parenthesis 7 close parenthesis")[$−5 ( 7 )$]

#solutionbox[
−35
]

#math.equation(block: true, alt: "−8 open parenthesis 6 close parenthesis")[$−8 ( 6 )$]

#math.equation(block: true, alt: "−18 open parenthesis −2 close parenthesis")[$−18 ( −2 )$]

#solutionbox[
36
]

#math.equation(block: true, alt: "−10 open parenthesis −6 close parenthesis")[$−10 ( −6 )$]

#math.equation(block: true, alt: "9 open parenthesis −7 close parenthesis")[$9 ( −7 )$]

#solutionbox[
−63
]

#math.equation(block: true, alt: "13 open parenthesis −5 close parenthesis")[$13 ( −5 )$]

#math.equation(block: true, alt: "−1 times 6")[$−1 · 6$]

#solutionbox[
−6
]

#math.equation(block: true, alt: "−1 times 3")[$−1 · 3$]

#math.equation(block: true, alt: "−1 open parenthesis −14 close parenthesis")[$−1 ( −14 )$]

#solutionbox[
14
]

#math.equation(block: true, alt: "−1 open parenthesis −19 close parenthesis")[$−1 ( −19 )$]

#strong[Divide Integers]

In the following exercises, divide.

#math.equation(block: true, alt: "−24 divided by 6")[$−24 ÷ 6$]

#solutionbox[
−4
]

#math.equation(block: true, alt: "−28 divided by 7")[$−28 ÷ 7$]

#math.equation(block: true, alt: "56 divided by open parenthesis −7 close parenthesis")[$56 ÷ ( −7 )$]

#solutionbox[
−8
]

#math.equation(block: true, alt: "35 divided by open parenthesis −7 close parenthesis")[$35 ÷ ( −7 )$]

#math.equation(block: true, alt: "−52 divided by open parenthesis −4 close parenthesis")[$−52 ÷ ( −4 )$]

#solutionbox[
13
]

#math.equation(block: true, alt: "−84 divided by open parenthesis −6 close parenthesis")[$−84 ÷ ( −6 )$]

#math.equation(block: true, alt: "−180 divided by 15")[$−180 ÷ 15$]

#solutionbox[
−12
]

#math.equation(block: true, alt: "−192 divided by 12")[$−192 ÷ 12$]

#math.equation(block: true, alt: "49 divided by open parenthesis −1 close parenthesis")[$49 ÷ ( −1 )$]

#solutionbox[
−49
]

#math.equation(block: true, alt: "62 divided by open parenthesis −1 close parenthesis")[$62 ÷ ( −1 )$]

#strong[Simplify Expressions with Integers]

In the following exercises, simplify each expression.

#math.equation(block: true, alt: "5 open parenthesis −6 close parenthesis plus 7 open parenthesis −2 close parenthesis −3")[$5 ( −6 ) + 7 ( −2 ) −3$]

#solutionbox[
−47
]

#math.equation(block: true, alt: "8 open parenthesis −4 close parenthesis plus 5 open parenthesis −4 close parenthesis −6")[$8 ( −4 ) + 5 ( −4 ) −6$]

#math.equation(block: true, alt: "−8 open parenthesis −2 close parenthesis −3 open parenthesis −9 close parenthesis")[$−8 ( −2 ) −3 ( −9 )$]

#solutionbox[
43
]

#math.equation(block: true, alt: "−7 open parenthesis −4 close parenthesis −5 open parenthesis −3 close parenthesis")[$−7 ( −4 ) −5 ( −3 )$]

#math.equation(block: true, alt: "open parenthesis −5 close parenthesis cubed")[$attach(( −5 ), t: 3)$]

#solutionbox[
−125
]

#math.equation(block: true, alt: "open parenthesis −4 close parenthesis cubed")[$attach(( −4 ), t: 3)$]

#math.equation(block: true, alt: "open parenthesis −2 close parenthesis to the power 6")[$attach(( −2 ), t: 6)$]

#solutionbox[
64
]

#math.equation(block: true, alt: "open parenthesis −3 close parenthesis to the power 5")[$attach(( −3 ), t: 5)$]

#math.equation(block: true, alt: "minus 4 squared")[$− 4^(2)$]

#solutionbox[
−16
]

#math.equation(block: true, alt: "minus 6 squared")[$− 6^(2)$]

#math.equation(block: true, alt: "−3 open parenthesis −5 close parenthesis open parenthesis 6 close parenthesis")[$−3 ( −5 ) ( 6 )$]

#solutionbox[
90
]

#math.equation(block: true, alt: "−4 open parenthesis −6 close parenthesis open parenthesis 3 close parenthesis")[$−4 ( −6 ) ( 3 )$]

#math.equation(block: true, alt: "−4 times 2 times 11")[$−4 · 2 · 11$]

#solutionbox[
−88
]

#math.equation(block: true, alt: "−5 times 3 times 10")[$−5 · 3 · 10$]

#math.equation(block: true, alt: "open parenthesis 8 minus 11 close parenthesis open parenthesis 9 minus 12 close parenthesis")[$( 8 − 11 ) ( 9 − 12 )$]

#solutionbox[
9
]

#math.equation(block: true, alt: "open parenthesis 6 minus 11 close parenthesis open parenthesis 8 minus 13 close parenthesis")[$( 6 − 11 ) ( 8 − 13 )$]

#math.equation(block: true, alt: "26 minus 3 open parenthesis 2 minus 7 close parenthesis")[$26 − 3 ( 2 − 7 )$]

#solutionbox[
41
]

#math.equation(block: true, alt: "23 minus 2 open parenthesis 4 minus 6 close parenthesis")[$23 − 2 ( 4 − 6 )$]

#math.equation(block: true, alt: "−10 open parenthesis −4 close parenthesis divided by open parenthesis −8 close parenthesis")[$−10 ( −4 ) ÷ ( −8 )$]

#solutionbox[
−5
]

#math.equation(block: true, alt: "−8 open parenthesis −6 close parenthesis divided by open parenthesis −4 close parenthesis")[$−8 ( −6 ) ÷ ( −4 )$]

#math.equation(block: true, alt: "65 divided by open parenthesis −5 close parenthesis plus open parenthesis −28 close parenthesis divided by open parenthesis −7 close parenthesis")[$65 ÷ ( −5 ) + ( −28 ) ÷ ( −7 )$]

#solutionbox[
−9
]

#math.equation(block: true, alt: "52 divided by open parenthesis −4 close parenthesis plus open parenthesis −32 close parenthesis divided by open parenthesis −8 close parenthesis")[$52 ÷ ( −4 ) + ( −32 ) ÷ ( −8 )$]

#math.equation(block: true, alt: "9 minus 2 [ 3 minus 8 open parenthesis −2 close parenthesis ]")[$9 − 2 [ 3 − 8 ( −2 ) ]$]

#solutionbox[
−29
]

#math.equation(block: true, alt: "11 minus 3 [ 7 minus 4 open parenthesis −2 close parenthesis ]")[$11 − 3 [ 7 − 4 ( −2 ) ]$]

#math.equation(block: true, alt: "open parenthesis −3 close parenthesis squared −24 divided by open parenthesis 8 minus 2 close parenthesis")[$attach(( −3 ), t: 2) −24 ÷ ( 8 − 2 )$]

#solutionbox[
5
]

#math.equation(block: true, alt: "open parenthesis −4 close parenthesis squared minus 32 divided by open parenthesis 12 minus 4 close parenthesis")[$attach(( −4 ), t: 2) − 32 ÷ ( 12 − 4 )$]

#strong[Evaluate Variable Expressions with Integers]

In the following exercises, evaluate each expression.

#math.equation(block: true, alt: "−2 x plus 17 when")[$−2 x + 17 #h(0.2em) "when" #h(0.2em)$]

+ ⓐ #math.equation(block: false, alt: "x equals 8")[$#h(0.2em) x = 8 #h(1em)$]
+ ⓑ #math.equation(block: false, alt: "x equals −8")[$#h(0.2em) x = −8$]

#solutionbox[
+ ⓐ 1
+ ⓑ 33
]

#math.equation(block: true, alt: "−5 y plus 14 when")[$−5 y + 14 #h(0.2em) "when" #h(0.2em)$]

+ ⓐ 
#math.equation(block: false, alt: "y equals 9")[$#h(0.2em) y = 9 #h(1em)$]
+ ⓑ 
#math.equation(block: false, alt: "y equals −9")[$#h(0.2em) y = −9$]

#math.equation(block: true, alt: "10 minus 3 m when")[$10 − 3 m #h(0.2em) "when" #h(0.2em)$]

+ ⓐ 
#math.equation(block: false, alt: "m equals 5")[$#h(0.2em) m = 5 #h(1em)$]
+ ⓑ 
#math.equation(block: false, alt: "m equals −5")[$#h(0.2em) m = −5$]

#solutionbox[
+ ⓐ −5
+ ⓑ 25
]

#math.equation(block: true, alt: "18 minus 4 n when")[$18 − 4 n #h(0.2em) "when" #h(0.2em)$]

+ ⓐ 
#math.equation(block: false, alt: "n equals 3")[$#h(0.2em) n = 3 #h(1em)$]
+ ⓑ 
#math.equation(block: false, alt: "n equals −3")[$#h(0.2em) n = −3$]

#math.equation(block: true, alt: "p squared minus 5 p plus 5 when p equals −1")[$p^(2) − 5 p + 5 #h(0.2em) "when" #h(0.2em) p = −1$]

#solutionbox[
11
]

#math.equation(block: false, alt: "q squared minus 2 q plus 9")[$q^(2) − 2 q + 9$] when #math.equation(block: false, alt: "q equals −2")[$q = −2$]

#math.equation(block: false, alt: "2 w squared minus 3 w plus 7")[$2 w^(2) − 3 w + 7$] when #math.equation(block: false, alt: "w equals −2")[$w = −2$]

#solutionbox[
21
]

#math.equation(block: false, alt: "3 u squared minus 4 u plus 5")[$3 u^(2) − 4 u + 5$] when #math.equation(block: false, alt: "u equals −3")[$u = −3$]

#math.equation(block: false, alt: "6 x minus 5 y plus 15")[$6 x − 5 y + 15$] when #math.equation(block: false, alt: "x equals 3")[$x = 3$] and #math.equation(block: false, alt: "y equals −1")[$y = −1$]

#solutionbox[
38
]

#math.equation(block: false, alt: "3 p minus 2 q plus 9")[$3 p − 2 q + 9$] when #math.equation(block: false, alt: "p equals 8")[$p = 8$] and #math.equation(block: false, alt: "q equals −2")[$q = −2$]

#math.equation(block: false, alt: "9 a minus 2 b minus 8")[$9 a − 2 b − 8$] when #math.equation(block: false, alt: "a equals −6")[$a = −6$] and #math.equation(block: false, alt: "b equals −3")[$b = −3$]

#solutionbox[
−56
]

#math.equation(block: false, alt: "7 m minus 4 n minus 2")[$7 m − 4 n − 2$] when #math.equation(block: false, alt: "m equals −4")[$m = −4$] and #math.equation(block: false, alt: "n equals −9")[$n = −9$]

#strong[Translate Word Phrases to Algebraic Expressions]

In the following exercises, translate to an algebraic expression and simplify if possible.

The product of #math.equation(block: false, alt: "−3")[$−3$] and 15

#solutionbox[
−3·15 = −45
]

The product of #math.equation(block: false, alt: "−4")[$−4$] and #math.equation(block: false, alt: "16")[$16$]

The quotient of #math.equation(block: false, alt: "−60")[$−60$] and #math.equation(block: false, alt: "−20")[$−20$]

#solutionbox[
−60 ÷ (−20) = 3
]

The quotient of #math.equation(block: false, alt: "−40")[$−40$] and #math.equation(block: false, alt: "−20")[$−20$]

The quotient of #math.equation(block: false, alt: "−6")[$−6$] and the sum of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$]

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

The quotient of #math.equation(block: false, alt: "−7")[$−7$] and the sum of #math.equation(block: false, alt: "m")[$m$] and #math.equation(block: false, alt: "n")[$n$]

The product of #math.equation(block: false, alt: "−10")[$−10$] and the difference of #math.equation(block: false, alt: "p and q")[$p #h(0.2em) "and" #h(0.2em) q$]

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

The product of #math.equation(block: false, alt: "−13")[$−13$] and the difference of #math.equation(block: false, alt: "c and d")[$c #h(0.2em) "and" #h(0.2em) d$]


==== Everyday Math

#strong[Stock market] Javier owns #math.equation(block: false, alt: "300")[$300$] shares of stock in one company. On Tuesday, the stock price dropped #math.equation(block: false, alt: "$12")[$"$12"$] per share. What was the total effect on Javier’s portfolio?

#solutionbox[
−\$3,600
]

#strong[Weight loss] In the first week of a diet program, eight women lost an average of #math.equation(block: false, alt: "3 pounds")[$"3 pounds"$] each. What was the total weight change for the eight women?


==== Writing Exercises

In your own words, state the rules for multiplying two integers.

#solutionbox[
Sample answer: Multiplying two integers with the same sign results in a positive product. Multiplying two integers with different signs results in a negative product.
]

In your own words, state the rules for dividing two integers.

Why is #math.equation(block: false, alt: "−2 to the power 4 not equal to open parenthesis −2 close parenthesis to the power 4 ?")[$−2^(4) ≠ attach(( −2 ), t: 4) ?$]

#solutionbox[
Sample answer: In the first expression the base is positive and after you raise it to the power you should take the opposite. Then in the second expression the base is negative so you simply raise it to the power.
]

Why is #math.equation(block: false, alt: "−4 squared not equal to open parenthesis −4 close parenthesis squared ?")[$−4^(2) ≠ attach(( −4 ), t: 2) ?$]


==== 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 math skills including multiplying and dividing integers, simplifying expressions, evaluating variable expressions, and translating word phrases to algebraic expressions.], alt: "A self-assessment chart for math skills including multiplying and dividing integers, simplifying expressions, evaluating variable expressions, and translating word phrases to algebraic expressions.", caption: none)

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?