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Simplify:

18 ( −4 ) ÷ ( −2 ) 3

9

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Simplify: −30 ÷ 2 + ( −3 ) ( −7 ) .

Solution

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

−30 ÷ 2 + ( −3 ) ( −7 )
Divide. −15 + ( −3 ) ( −7 )
Multiply. −15 + 21
Add. 6
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Simplify:

−27 ÷ 3 + ( −5 ) ( −6 )

21

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Simplify:

−32 ÷ 4 + ( −2 ) ( −7 )

6

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Evaluate variable expressions with integers

Now we can evaluate 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.

Evaluate 2 x 2 3 x + 8 when x = −4 .

Solution

.
. .
Simplify exponents. .
Multiply. .
Subtract. .
Add. .

Keep in mind that when we substitute −4 for x , we use parentheses to show the multiplication. Without parentheses, it would look like 2 · −4 2 3 · −4 + 8 .

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Evaluate:

3 x 2 2 x + 6 when x = −3

39

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Evaluate:

4 x 2 x 5 when x = −2

13

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Evaluate 3 x + 4 y 6 when x = −1 and y = 2 .

Solution

.
Substitute x = −1 and y = 2 . .
Multiply. .
Simplify. .
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Evaluate:

7 x + 6 y 12 when x = −2 and y = 3

−8

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Evaluate:

8 x 6 y + 13 when x = −3 and y = −5

19

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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 product and for division is quotient .

Translate to an algebraic expression and simplify if possible: the product of −2 and 14 .

Solution

The word product tells us to multiply.

the product of −2 and 14
Translate. ( −2 ) ( 14 )
Simplify. −28
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Translate to an algebraic expression and simplify if possible:

the product of −5 and 12

−5 (12) = −60

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Translate to an algebraic expression and simplify if possible:

the product of 8 and −13

8 (−13) = −104

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Translate to an algebraic expression and simplify if possible: the quotient of −56 and −7 .

Solution

The word quotient tells us to divide.

the quotient of −56 and −7
Translate. −56 ÷ ( −7 )
Simplify. 8
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Translate to an algebraic expression and simplify if possible:

the quotient of −63 and −9

−63 ÷ −9 = 7

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Translate to an algebraic expression and simplify if possible:

the quotient of −72 and −9

−72 ÷ −9 = 8

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Key concepts

  • Multiplication and Division of Signed Numbers
    Same signs Product
    •Two positives
    •Two negatives
    Positive
    Positive
    Different signs Product
    •Positive • negative
    •Negative • positive
    Negative
    Negative
  • Multiplication by −1
    • Multiplying a number by −1 gives its opposite: −1 a = a
  • Division by −1
    • Dividing a number by −1 gives its opposite: a ÷ ( −1 ) = −a

Practice makes perfect

Multiply Integers

In the following exercises, multiply each pair of integers.

Divide Integers

In the following exercises, divide.

Simplify Expressions with Integers

In the following exercises, simplify each expression.

5 ( −6 ) + 7 ( −2 ) −3

−47

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8 ( −4 ) + 5 ( −4 ) −6

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−8 ( −2 ) −3 ( −9 )

43

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−7 ( −4 ) −5 ( −3 )

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( 8 11 ) ( 9 12 )

9

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( 6 11 ) ( 8 13 )

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−10 ( −4 ) ÷ ( −8 )

−5

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−8 ( −6 ) ÷ ( −4 )

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65 ÷ ( −5 ) + ( −28 ) ÷ ( −7 )

−9

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52 ÷ ( −4 ) + ( −32 ) ÷ ( −8 )

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9 2 [ 3 8 ( −2 ) ]

−29

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11 3 [ 7 4 ( −2 ) ]

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( −3 ) 2 −24 ÷ ( 8 2 )

5

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( −4 ) 2 32 ÷ ( 12 4 )

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Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

−2 x + 17 when

  1. x = 8
  2. x = −8

  1. 1
  2. 33

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−5 y + 14 when

  1. y = 9
  2. y = −9

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10 3 m when

  1. m = 5
  2. m = −5

  1. −5
  2. 25

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18 4 n when

  1. n = 3
  2. n = −3

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p 2 5 p + 5 when p = −1

8

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q 2 2 q + 9 when q = −2

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2 w 2 3 w + 7 when w = −2

21

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3 u 2 4 u + 5 when u = −3

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6 x 5 y + 15 when x = 3 and y = −1

38

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3 p 2 q + 9 when p = 8 and q = −2

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9 a 2 b 8 when a = −6 and b = −3

−56

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7 m 4 n 2 when m = −4 and n = −9

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Translate Word Phrases to Algebraic Expressions

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

The product of −3 and 15

−3·15 = −45

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The product of −4 and 16

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The quotient of −60 and −20

−60 ÷ (−20) = 3

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The quotient of −40 and −20

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The quotient of −6 and the sum of a and b

−6 a + b

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The quotient of −7 and the sum of m and n

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The product of −10 and the difference of p and q

−10 ( p q )

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The product of −13 and the difference of c and d



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Everyday math

Stock market Javier owns 300 shares of stock in one company. On Tuesday, the stock price dropped $12 per share. What was the total effect on Javier’s portfolio?

−$3,600

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Weight loss In the first week of a diet program, eight women lost an average of 3 pounds each. What was the total weight change for the eight women?

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Writing exercises

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

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.

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

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Why is −2 4 ( −2 ) 4 ?

Sample answer: In one expression the base is positive and then we take the opposite, but in the other the base is negative.

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Why is −4 2 ( −4 ) 2 ?

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Self check

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

.

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?

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
Difference between extinct and extici spicies
Amanpreet Reply
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
Beverly Reply
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
Mapo Reply
The formulation of new memories is sometimes called ________, and the process of bringing up old memories is called ________.
Mapo Reply
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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