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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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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
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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