<< Chapter < Page Chapter >> Page >

( 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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Practice set b

Write each of the following without exponents.

( 4 a ) 3

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

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Practice set d

Use the order of operations to simplify the following.

1 4 + ( 2 2 + 4 ) 2 ÷ 2 3

9

Got questions? Get instant answers now!

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

8

Got questions? Get instant answers now!

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

3

Got questions? Get instant answers now!

Exercises

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

3 squared times y to the fifth

Got questions? Get instant answers now!

a cubed minus ( b + 7 ) squared

a 3 ( b + 7 ) 2

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

2 ( 3 4 ) x 2 y 5

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

7 x 2 ( a + 8 ) 2

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

4 x 4 x 4 x 4 x 4 x

( 4 x ) 5 or 4 5 x 5

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

( 7 ) 4 a 5 b 5

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

( z + w ) 3 ( z w ) 2

Got questions? Get instant answers now!

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

3 x 3 y 2 ( x + 1 ) 3

Got questions? Get instant answers now!

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

8 x 3 y 2

8 · x · x · x · y · y

Got questions? Get instant answers now!

( 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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Select x = 2. Then, 196 28.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

zero

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

( 4 + 3 ) 2 + 1 ÷ ( 2 5 )

Got questions? Get instant answers now!

( 2 4 + 2 5 2 3 5 ) 2 ÷ 4 2

4

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

71

Got questions? Get instant answers now!

( 1 + 6 ) 2 + 2 19

51 19

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

5

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

1070 11 or 97. 27 ¯

Got questions? Get instant answers now!

Exercises for review

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

Got questions? Get instant answers now!

( [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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

x y

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
hey
Giriraj
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
ya I also want to know the raman spectra
Bhagvanji
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
Please keep in mind that it's not allowed to promote any social groups (whatsapp, facebook, etc...), exchange phone numbers, email addresses or ask for personal information on QuizOver's platform.
QuizOver Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask