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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The symbols, notations, and properties of numbers that form the basis of algebra, as well as exponents and the rules of exponents, are introduced in this chapter. Each property of real numbers and the rules of exponents are expressed both symbolically and literally. Literal explanations are included because symbolic explanations alone may be difficult for a student to interpret.This module contains the exercise supplement for the chapter "Basic Properties of Real Numbers".

Exercise supplement

Symbols and notations ( [link] )

For the following problems, simplify the expressions.

9 ( 4 2 ) + 6 ( 8 + 2 ) 3 ( 1 + 4 )

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6 [ 1 + 8 ( 7 + 2 ) ]

438

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( 4 + 17 + 1 ) + 4 14 1

2

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( 4 + 5 ) ( 4 + 6 ) ( 4 + 7 )

79

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8 ( 2 12 ÷ 13 ) + 2 5 11 [ 1 + 4 ( 1 + 2 ) ]

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3 4 + 1 12 ( 3 4 1 2 )

37 48

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88 11 + 99 9 + 1 54 9 22 11

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8 6 2 + 9 9 3 10 4 5

43

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For the following problems, write the appropriate relation symbol ( = , < , > ) in place of the .

9 [ 4 + 3 ( 8 ) ] 6 [ 1 + 8 ( 5 ) ]

252 > 246

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3 ( 1.06 + 2.11 ) 4 ( 11.01 9.06 )

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For the following problems, state whether the letters or symbols are the same or different.

Represent the sum of c and d two different ways.

c + d ; d + c

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For the following problems, use algebraic notataion.

62 divided by f

62 f or 62 ÷ f

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6 times x , minus 2

6 x 2

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x + 1 divided by x 3

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y + 11 divided by y + 10 , minus 12

( y + 11 ) ÷ ( y + 10 ) 12 or y + 11 y + 10 12

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The real number line and the real numbers ( [link] )

Is every natural number a whole number?

yes

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Is every rational number a real number?

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For the following problems, locate the numbers on a number line by placing a point at their (approximate) position.

Draw a number line that extends from 10 to 20. Place a point at all odd integers.

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Draw a number line that extends from 10 to 10 . Place a point at all negative odd integers and at all even positive integers.

A number line with arrows on each end, labeled from negative ten to ten in increments of two. There are closed circles at negative nine, negative seven, negative five, negative three, negative one, two, four, six, eight, and ten.

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Draw a number line that extends from 5 to 10 . Place a point at all integers that are greater then or equal to 2 but strictly less than 5.

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Draw a number line that extends from 10 to 10 . Place a point at all real numbers that are strictly greater than 8 but less than or equal to 7.

A number line with arrows on each end, labeled from negative ten to ten in increments of two. There is a closed circle at seven and an open circle at negative eight, with a black shaded line connecting the two circles.

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Draw a number line that extends from 10 to 10 . Place a point at all real numbers between and including 6 and 4.

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For the following problems, write the appropriate relation symbol ( = , < , > ).

8 5

8 < 5

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Is there a smallest two digit integer? If so, what is it?

yes, 99

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Is there a smallest two digit real number? If so, what is it?

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For the following problems, what integers can replace x so that the statements are true?

4 x 7

4 , 5 , 6 , or 7

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3 < x 2

2 , 1 , 0 , 1 , or 2

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The temperature today in Los Angeles was eighty-two degrees. Represent this temperature by real number.

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The temperature today in Marbelhead was six degrees below zero. Represent this temperature by real number.

6 °

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On the number line, how many units between 3 and 2?

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On the number line, how many units between 4 and 0?

4

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Properties of the real numbers ( [link] )

a + b = b + a is an illustration of the property of addition.

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s t = t s is an illustration of the __________ property of __________.

commutative, multiplication

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Use the commutative properties of addition and multiplication to write equivalent expressions for the following problems.

2 ( a 1 )

( a 1 ) 2

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( 6 ) ( 9 ) ( 2 )

( 9 ) ( 6 ) ( 2 ) or  ( 9 ) ( 2 ) ( 6 ) or ( 6 ) ( 2 ) ( 9 ) or ( 2 ) ( 9 ) ( 6 )

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

3 ( x + 2 ) 5 ( x 1 ) 0 ( x + 6 )

0

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8 b ( a 6 ) 9 a ( a 4 )

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

2 g ( 4 h + 2 k )

8 g h + 4 g k

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3 y ( 2 x + 4 z + 5 w )

6 x y + 12 y z + 15 w y

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( x + y ) ( 4 a + 3 b )

4 a x + 3 b x + 4 a y + 3 b y

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Exponents ( [link] )

For the following problems, write the expressions using exponential notation.

( a + 2 b ) squared minus ( a + 3 b ) to the fourth.

( a + 2 b ) 2 ( a + 3 b ) 4

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x cubed plus 2 times ( y x ) to the seventh.

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( 8 ) ( 8 ) ( 8 ) ( 8 ) x x x y y y y y

( 8 ) 4 x 3 y 5

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( x 9 ) ( x 9 ) + ( 3 x + 1 ) ( 3 x + 1 ) ( 3 x + 1 )

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2 z z y z y y y + 7 z z y z ( a 6 ) 2 ( a 6 )

2 y 4 z 3 + 7 y z 3 ( a 6 ) 3

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For the following problems, expand the terms so that no exponents appear.

( 4 b ) 2

4 b · 4 b

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( 6 a 2 ) 3 ( 5 c 4 ) 2

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( x 3 + 7 ) 2 ( y 2 3 ) 3 ( z + 10 )

( x x x + 7 ) ( x x x + 7 ) ( y y 3 ) ( y y 3 ) ( y y 3 ) ( z + 10 )

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Choose values for a and b to show that

  1. ( a + b ) 2 is not always equal to a 2 + b 2 .
  2. ( a + b ) 2 may be equal to a 2 + b 2 .
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Choose value for x to show that

  1. ( 4 x ) 2 is not always equal to 4 x 2 .
  2. ( 4 x ) 2 may be equal to 4 x 2 .

(a) any value except zero

(b) only zero

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Rules of exponents ( [link] ) - the power rules for exponents ( [link] )

Simplify the following problems.

1 8 + 0 10 + 3 2 ( 4 2 + 2 3 )

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12 2 + 0.3 ( 11 ) 2

180.3

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6 2 + 3 2 2 2 + 1 + ( 1 + 4 ) 2 2 3 1 4 2 5 4 2

10

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4 a 3 b 2 c 8 3 a b 2 c 0

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( 6 x 4 y 10 ) ( x y 3 )

6 x 5 y 13

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( 3 x y z 2 ) ( 2 x 2 y 3 ) ( 4 x 2 y 2 z 4 )

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( 3 4 x 8 y 6 z 0 a 10 b 15 ) 2

9 16 x 16 y 12 a 20 b 30

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14 a 4 b 6 c 7 2 a b 3 c 2

7 a 3 b 3 c 5

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a 3 b 7 a 9 b 6 a 5 b 10

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( x 4 y 6 z 10 ) 4 ( x y 5 z 7 ) 3

x 13 y 9 z 19

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( 2 x 1 ) 13 ( 2 x + 5 ) 5 ( 2 x 1 ) 10 ( 2 x + 5 )

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( 3 x 2 4 y 3 ) 2

9 x 4 16 y 6

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( x + y ) 9 ( x y ) 4 ( x + y ) 3

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6 b 2 n + 7 8 b 5 n + 2

48 b 7 n + 9

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18 x 4 n + 9 2 x 2 n + 1

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( x 5 t y 4 r ) 7

x 35 t y 28 r

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( a 2 n b 3 m c 4 p ) 6 r

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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
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
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
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
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
Prasenjit
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
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
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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