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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

Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
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
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
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what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
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Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
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many many of nanotubes
Porter
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what is the function of carbon nanotubes?
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I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
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AMJAD
what is the application of nanotechnology?
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In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
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after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
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name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
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silver nanoparticles could handle the job?
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not now but maybe in future only AgNP maybe any other nanomaterials
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I'm interested in Nanotube
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this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
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Prasenjit 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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