<< 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

what's Thermochemistry
rhoda Reply
the study of the heat energy which is associated with chemical reactions
Kaddija
How was CH4 and o2 was able to produce (Co2)and (H2o
Edafe Reply
explain please
Victory
First twenty elements with their valences
Martine Reply
what is chemistry
asue Reply
what is atom
asue
what is the best way to define periodic table for jamb
Damilola Reply
what is the change of matter from one state to another
Elijah Reply
what is isolation of organic compounds
IKyernum Reply
what is atomic radius
ThankGod Reply
Read Chapter 6, section 5
Dr
Read Chapter 6, section 5
Kareem
Atomic radius is the radius of the atom and is also called the orbital radius
Kareem
atomic radius is the distance between the nucleus of an atom and its valence shell
Amos
Read Chapter 6, section 5
paulino
Bohr's model of the theory atom
Ayom Reply
is there a question?
Dr
when a gas is compressed why it becomes hot?
ATOMIC
It has no oxygen then
Goldyei
read the chapter on thermochemistry...the sections on "PV" work and the First Law of Thermodynamics should help..
Dr
Which element react with water
Mukthar Reply
Mgo
Ibeh
an increase in the pressure of a gas results in the decrease of its
Valentina Reply
definition of the periodic table
Cosmos Reply
What is the lkenes
Da Reply
what were atoms composed of?
Moses Reply
what is chemistry
Imoh Reply
what is chemistry
Damilola
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




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