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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

how does the planets on our solar system orbit
cheten Reply
how many Messier objects are there in space
satish Reply
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Richard Reply
what are astronomy
Issan Reply
Astronomy (from Ancient Greek ἀστρονομία (astronomía) 'science that studies the laws of the stars') is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution.
Rafael
vjuvu
Elgoog
what is big bang theory?
Rosemary
what type of activity astronomer do?
Rosemary
No
Richard
the big bang theory is a theory which states that all matter was compressed together in one place the matter got so unstable it exploded releasing All its contents in the form of hydrogen
Roaul
I want to be an astronomer. That's my dream
Astrit
Who named the the whole galaxy?
Shola Reply
solar Univers
GPOWER
what is space
Richard
what is the dark matter
Richard
what are the factors upon which the atmosphere is stratified
Nicholas Reply
is the big bang the sun
Folakemi Reply
no
Sokak
bigbang is the beginning of the universe
Sokak
but thats just a theory
Sokak
nothing will happen, don't worry brother.
Vansh
what does comet means
GANGAIN Reply
these are Rocky substances between mars and jupiter
GANGAIN
Comets are cosmic snowballs of frozen gases , rock and dust that orbit the sun. They are mostly found between the orbits of Venus and Mercury.
Aarya
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John
hi
John
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John
r u there
John
hey can anyone guide me abt international astronomy olympiad
sahil
how can we learn right and true ?
Govinda Reply
why the moon is always appear in an elliptical shape
Gatjuol Reply
Because when astroid hit the Earth then a piece of elliptical shape of the earth was separated which is now called moon.
Hemen
what's see level?
lidiya Reply
Did you mean eye sight or sea level
Minal
oh sorry it's sea level
lidiya
according to the theory of astronomers why the moon is always appear in an elliptical orbit?
Gatjuol
hi !!! I am new in astronomy.... I have so many questions in mind .... all of scientists of the word they just give opinion only. but they never think true or false ... i respect all of them... I believes whole universe depending on true ...থিউরি
Govinda
hello
Jackson
hi
Elyana
we're all stars and galaxies a part of sun. how can science prove thx with respect old ancient times picture or books..or anything with respect to present time .but we r a part of that universe
w astronomy and cosmology!
Michele
another theory of universe except big ban
Albash Reply
how was universe born
Asmit Reply
there many theory to born universe but what is the reality of big bang theory to born universe
Asmit
what is the exact value of π?
Nagalakshmi
by big bang
universal
there are many theories regarding this it's on you believe any theory that you think is true ex. eternal inflation theory, oscillation model theory, multiple universe theory the big bang theory etc.
Aarya
I think after Big Bang!
Michele
from where on earth could u observe all the stars during the during the course of an year
Karuna Reply
I think it couldn't possible on earth
Nagalakshmi
in this time i don't Know
Michele
is that so. the question was in the end of this chapter
Karuna
in theory, you could see them all from the equator (though over the course of a year, not at pne time). stars are measured in "declination", which is how far N or S of the equator (90* to -90*). Polaris is the North star, and is ALMOST 90* (+89*). So it would just barely creep over the horizon.
Christopher
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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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