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( 8 x ) 3 means ( 8 x ) ( 8 x ) ( 8 x ) since the parentheses indicate that the exponent 3 is directly connected to the factor 8 x . Remember that the grouping symbols (   ) indicate that the quantities inside are to be considered as one single number.

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34 ( a + 1 ) 2 means 34 ( a + 1 ) ( a + 1 ) since the exponent 2 applies only to the factor ( a + 1 ) .

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Practice set b

Write each of the following without exponents.

( 4 a ) 3

( 4 a ) ( 4 a ) ( 4 a )

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Sample set c

Select a number to show that ( 2 x ) 2 is not always equal to 2 x 2 .

Suppose we choose x to be 5. Consider both ( 2 x ) 2 and 2 x 2 .

( 2 x ) 2 2 x 2 ( 2 5 ) 2 2 5 2 ( 10 ) 2 2 25 100 50

Notice that ( 2 x ) 2 = 2 x 2 only when x = 0 .

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Practice set c

Select a number to show that ( 5 x ) 2 is not always equal to 5 x 2 .

Select x = 3 . Then ( 5 3 ) 2 = ( 15 ) 2 = 225 , but 5 3 2 = 5 9 = 45 .     225 45 .

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Reading exponential notation

In x n ,

Base

x is the base

Exponent

n is the exponent

Power

The number represented by x n is called a power .

x To the n Th power

The term x n is read as " x to the n th power," or more simply as " x to the n th."

x Squared and x Cubed

The symbol x 2 is often read as " x squared," and x 3 is often read as " x cubed." A natural question is "Why are geometric terms appearing in the exponent expression?" The answer for x 3 is this: x 3 means x x x . In geometry, the volume of a rectangular box is found by multiplying the length by the width by the depth. A cube has the same length on each side. If we represent this length by the letter x then the volume of the cube is x x x , which, of course, is described by x 3 . (Can you think of why x 2 is read as x squared?)

Cube with
length = x
width = x
depth = x
Volume = x x x = x 3

A cube with length of side equal to x.

The order of operations

In Section [link] we were introduced to the order of operations. It was noted that we would insert another operation before multiplication and division. We can do that now.

    The order of operations

  1. Perform all operations inside grouping symbols beginning with the innermost set.
  2. Perform all exponential operations as you come to them, moving left-to-right.
  3. Perform all multiplications and divisions as you come to them, moving left-to-right.
  4. Perform all additions and subtractions as you come to them, moving left-to-right.

Sample set d

Use the order of operations to simplify each of the following.

5 2 + 3 2 + 10 = 25 + 9 + 10 = 44

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2 2 + ( 5 ) ( 8 ) 1 = 4 + ( 5 ) ( 8 ) 1 = 4 + 40 1 = 43

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7 6 4 2 + 1 5 = 7 6 16 + 1 = 42 16 + 1 = 27

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( 2 + 3 ) 3 + 7 2 3 ( 4 + 1 ) 2 = ( 5 ) 3 + 7 2 3 ( 5 ) 2 = 125 + 49 3 ( 25 ) = 125 + 49 75 = 99

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[ 4 ( 6 + 2 ) 3 ] 2 = [ 4 ( 8 ) 3 ] 2 = [ 4 ( 512 ) ] 2 = [ 2048 ] 2 = 4 , 194 , 304

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6 ( 3 2 + 2 2 ) + 4 2 = 6 ( 9 + 4 ) + 4 2 = 6 ( 13 ) + 4 2 = 6 ( 13 ) + 16 = 78 + 16 = 94

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6 2 + 2 2 4 2 + 6 2 2 + 1 3 + 8 2 10 2 ( 19 ) ( 5 ) = 36 + 4 16 + 6 4 + 1 + 64 100 95 = 36 + 4 16 + 24 + 1 + 64 100 95 = 40 40 + 65 5 = 1 + 13 = 14

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Practice set d

Use the order of operations to simplify the following.

1 4 + ( 2 2 + 4 ) 2 ÷ 2 3

9

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[ 6 ( 10 2 3 ) ] 2 10 2 6 2

8

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5 2 + 6 2 10 1 + 4 2 + 0 4 0 5 7 2 6 2 3

3

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Exercises

For the following problems, write each of the quantities using exponential notation.

3 squared times y to the fifth

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a cubed minus ( b + 7 ) squared

a 3 ( b + 7 ) 2

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( 21 x ) cubed plus ( x + 5 ) to the seventh

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2 3 3 3 3 x x y y y y y

2 ( 3 4 ) x 2 y 5

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2 2 5 6 6 6 x y y z z z w w w w

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7 x x ( a + 8 ) ( a + 8 )

7 x 2 ( a + 8 ) 2

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10 x y y ( c + 5 ) ( c + 5 ) ( c + 5 )

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4 x 4 x 4 x 4 x 4 x

( 4 x ) 5 or 4 5 x 5

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( 9 a ) ( 9 a ) ( 9 a ) ( 9 a )

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( 7 ) ( 7 ) ( 7 ) a a b b b a ( 7 ) b a a b

( 7 ) 4 a 5 b 5

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( a 10 ) ( a 10 ) ( a + 10 )

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( z + w ) ( z + w ) ( z + w ) ( z w ) ( z w )

( z + w ) 3 ( z w ) 2

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3 x y x x y ( x + 1 ) ( x + 1 ) ( x + 1 )

3 x 3 y 2 ( x + 1 ) 3

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

8 x 3 y 2

8 · x · x · x · y · y

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( 9 a 3 b 2 ) 3

( 9 a a a b b ) ( 9 a a a b b ) ( 9 a a a b b ) or 9 · 9 · 9 a a a a a a a a a b b b b b b

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10 a 3 b 2 ( 3 c ) 2

10 a a a b b ( 3 c ) ( 3 c )  or 10 · 3 · 3 a a a b b c c

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( a + 10 ) 2 ( a 2 + 10 ) 2

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( x 2 y 2 ) ( x 2 + y 2 )

( x x y y ) ( x x + y y )

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For the following problems, select a number (or numbers) to show that

( 5 x ) 2 is not generally equal to 5 x 2 .

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( 7 x ) 2 is not generally equal to 7 x 2 .

Select x = 2. Then, 196 28.

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( a + b ) 2 is not generally equal to a 2 + b 2 .

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For what real number is ( 6 a ) 2 equal to 6 a 2 ?

zero

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For what real numbers, a and b , is ( a + b ) 2 equal to a 2 + b 2 ?

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Use the order of operations to simplify the quantities for the following problems.

( 4 + 3 ) 2 + 1 ÷ ( 2 5 )

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( 2 4 + 2 5 2 3 5 ) 2 ÷ 4 2

4

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1 6 + 0 8 + 5 2 ( 2 + 8 ) 3

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( 7 ) ( 16 ) 9 2 + 4 ( 1 1 + 3 2 )

71

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( 1 + 6 ) 2 + 2 19

51 19

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6 2 1 5 + 4 3 + ( 2 ) ( 3 ) 10

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5 [ 8 2 9 ( 6 ) ] 2 5 7 + 7 2 4 2 2 4 5

5

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( 2 + 1 ) 3 + 2 3 + 1 3 6 2 15 2 [ 2 ( 5 ) ] 2 5 5 2

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6 3 2 10 2 2 2 + 18 ( 2 3 + 7 2 ) 2 ( 19 ) 3 3

1070 11 or 97. 27 ¯

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Exercises for review

( [link] ) Use algebraic notation to write the statement "a number divided by eight, plus five, is equal to ten."

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( [link] ) Draw a number line that extends from 5 to 5 and place points at all real numbers that are strictly greater than 3 but less than or equal to 2.

A number line with arrows on each end, labeled from negative five to five in increments of one. There is a closed circle at two, and an open circle at negative three. These circles are connected by a black line.

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( [link] ) Is every integer a whole number?

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( [link] ) Use the commutative property of multiplication to write a number equal to the number y x .

x y

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( [link] ) Use the distributive property to expand 3 ( x + 6 ) .

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Questions & Answers

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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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
Do you know which machine is used to that process?
s.
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
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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