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( 9 y ) 4 = 9 ( y 4 ) Both represent the same product .

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Practice set b

Fill in the ( ) to make each statement true. Use the associative properties.

( 9 + 2 ) + 5 = 9 + ( )

2 + 5

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x + ( 5 + y ) = ( ) + y

x + 5

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( 11 a ) 6 = 11 ( )

a 6

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[ ( 7 m 2 ) ( m + 3 ) ] ( m + 4 ) = ( 7 m 2 ) [ ( ) ( ) ]

( m + 3 ) ( m + 4 )

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Sample set c

Simplify (rearrange into a simpler form): 5 x 6 b 8 a c 4 .

According to the commutative property of multiplication, we can make a series of consecutive switches and get all the numbers together and all the letters together.

5 6 8 4 x b a c 960 x b a c Multiply the numbers . 960 a b c x By convention, we will, when possible, write all letters in alphabetical order .

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Practice set c

Simplify each of the following quantities.

6 b 8 a c z 4 5

960 a b c z

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4 p 6 q r 3 ( a + b )

72 p q r ( a + b )

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The distributive properties

When we were first introduced to multiplication we saw that it was developed as a description for repeated addition.

4 + 4 + 4 = 3 4

Notice that there are three 4’s, that is, 4 appears 3 times . Hence, 3 times 4.
We know that algebra is generalized arithmetic. We can now make an important generalization.

When a number a is added repeatedly n times, we have
a + a + a + + a a appears n times
Then, using multiplication as a description for repeated addition, we can replace
a + a + a + + a n times with n a

For example:

x + x + x + x can be written as 4 x since x is repeatedly added 4 times.

x + x + x + x = 4 x

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r + r can be written as 2 r since r is repeatedly added 2 times.

r + r = 2 r

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The distributive property involves both multiplication and addition. Let’s rewrite 4 ( a + b ) . We proceed by reading 4 ( a + b ) as a multiplication: 4 times the quantity ( a + b ) . This directs us to write

4 ( a + b ) = ( a + b ) + ( a + b ) + ( a + b ) + ( a + b ) = a + b + a + b + a + b + a + b

Now we use the commutative property of addition to collect all the a ' s together and all the b ' s together.

4 ( a + b ) = a + a + a + a 4 a ' s + b + b + b + b 4 b ' s

Now, using multiplication as a description for repeated addition, we have

4 ( a + b ) = 4 a + 4 b

We have distributed the 4 over the sum to both a and b .

The product of four and the expression, a plus b, is equal to four a plus four b. The distributive property is shown by the arrows from four to each term of expression a plus b in the product.

The distributive property

a ( b + c ) = a b + a c ( b + c ) a = a b + a c

The distributive property is useful when we cannot or do not wish to perform operations inside parentheses.

Sample set d

Use the distributive property to rewrite each of the following quantities.

Practice set d

What property of real numbers justifies
a ( b + c ) = ( b + c ) a ?

the commutative property of multiplication

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Use the distributive property to rewrite each of the following quantities.

The identity properties

Additive identity

The number 0 is called the additive identity since when it is added to any real number, it preserves the identity of that number. Zero is the only additive identity.
For example, 6 + 0 = 6 .

Multiplicative identity

The number 1 is called the multiplicative identity since when it multiplies any real number, it preserves the identity of that number. One is the only multiplicative identity.
For example 6 1 = 6 .

We summarize the identity properties as follows.

ADDITIVE IDENTITY PROPERTY MULTIPLICATIVE IDENTITY PROPERTY If a is a real number, then If a is a real number, then a + 0 = a and 0 + a = a a 1 = a and 1 a = a

The inverse properties

Additive inverses

When two numbers are added together and the result is the additive identity, 0, the numbers are called additive inverses of each other. For example, when 3 is added to 3 the result is 0, that is, 3 + ( 3 ) = 0 . The numbers 3 and 3 are additive inverses of each other.

Multiplicative inverses

When two numbers are multiplied together and the result is the multiplicative identity, 1, the numbers are called multiplicative inverses of each other. For example, when 6 and 1 6 are multiplied together, the result is 1, that is, 6 1 6 = 1 . The numbers 6 and 1 6 are multiplicative inverses of each other.

We summarize the inverse properties as follows.

    The inverse properties

  1. If a is any real number, then there is a unique real number a , such that
    a + ( a ) = 0 and a + a = 0
    The numbers a and a are called additive inverses of each other.
  2. If a is any nonzero real number, then there is a unique real number 1 a such that
    a 1 a = 1 and 1 a a = 1
    The numbers a and 1 a are called multiplicative inverses of each other.

Expanding quantities

When we perform operations such as 6 ( a + 3 ) = 6 a + 18 , we say we are expanding the quantity 6 ( a + 3 ) .

Exercises

Use the commutative property of addition and multiplication to write expressions for an equal number for the following problems. You need not perform any calculations.

( x + 16 ) ( a + 7 )

( a + 7 ) ( x + 16 )

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5 ( 6 h + 1 )

( 6 h + 1 ) 5

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k ( 10 a b )

( 10 a b ) k

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( 16 ) ( 4 )

( 4 ) ( 16 )

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Simplify using the commutative property of multiplication for the following problems. You need not use the distributive property.

1 u 3 r 2 z 5 m 1 n

30 m n r u z

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6 d 4 e 1 f 2 ( g + 2 h )

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( 1 2 ) d ( 1 4 ) e ( 1 2 ) a

1 16 a d e

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3 ( a + 6 ) 2 ( a 9 ) 6 b

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1 ( x + 2 y ) ( 6 + z ) 9 ( 3 x + 5 y )

9 ( x + 2 y ) ( 6 + z ) ( 3 x + 5 y )

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For the following problems, use the distributive property to expand the quantities.

z ( x + 9 w )

x z + 9 w z

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( 8 + 2 f ) g

8 g + 2 f g

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15 x ( 2 y + 3 z )

30 x y + 45 x z

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z ( x + y + m )

x z + y z + m z

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( x + 10 ) ( a + b + c )

a x + b x + c x + 10 a + 10 b + 10 c

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Use a calculator. 21.5 ( 16.2 a + 3.8 b + 0.7 c )

348.3 a + 81.7 b + 15.05 c

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2 z t ( L m + 8 k )

2 L m z t + 16 k z t

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Exercises for review

( [link] ) Find the value of 4 2 + 5 ( 2 4 6 ÷ 3 ) 2 5 .

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( [link] ) Is the statement 3 ( 5 3 3 5 ) + 6 2 3 4 < 0 true or false?

false

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( [link] ) Draw a number line that extends from 2 to 2 and place points at all integers between and including 2 and 3.

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( [link] ) Replace the with the appropriate relation symbol ( < , > ) . 7 3 .

<

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( [link] ) What whole numbers can replace x so that the statement 2 x < 2 is true?

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Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
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
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
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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