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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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir 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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