- Exponential Notation
- Reading Exponential Notation
- The Order of Operations
In Section (Reference) we were reminded that multiplication is a description for repeated addition. A natural question is “Is there a description for repeated multiplication?” The answer is yes. The notation that describes repeated multiplication is exponential notation.
In multiplication, the numbers being multiplied together are called factors. In repeated multiplication, all the factors are the same. In nonrepeated multiplication, none of the factors are the same.
For example,
18⋅18⋅18⋅18
Repeated multiplication of 18. All four factors, 18, are the same.
x⋅x⋅x⋅x⋅x
Repeated multiplication of x. All five factors, x,are the same.
3⋅7⋅a
Nonrepeated multiplication. None of the factors are the same.
18⋅18⋅18⋅18
Repeated multiplication of 18. All four factors, 18, are the same.
x⋅x⋅x⋅x⋅x
Repeated multiplication of x. All five factors, x,are the same.
3⋅7⋅a
Nonrepeated multiplication. None of the factors are the same.
Exponential notation is used to show repeated multiplication of the same factor. The notation consists of using a superscript on the factor that is repeated. The superscript is called an exponent.
If
x
x
is any real number and
n
n
is a natural number, then
x
n
=
x⋅x⋅x⋅...⋅x
︸
n factors of x
x
n
=
x⋅x⋅x⋅...⋅x
︸
n factors of x
An exponent records the number of identical factors in a multiplication.
Note that the definition for exponential notation only has meaning for natural number exponents. We will extend this notation to include other numbers as exponents later.
7⋅7⋅7⋅7⋅7⋅7=
7
6
7⋅7⋅7⋅7⋅7⋅7=
7
6
.
The repeated factor is 7. The exponent 6 records the fact that 7 appears 6 times in the multiplication.
x⋅x⋅x⋅x=
x
4
x⋅x⋅x⋅x=
x
4
.
The repeated factor is
x
x
. The exponent 4 records the fact that
x
x
appears 4 times in the multiplication.
(2y)(2y)(2y)=
(2y)
3
(2y)(2y)(2y)=
(2y)
3
.
The repeated factor is
2y
2y
. The exponent 3 records the fact that the factor
2y
2y
appears 3 times in the multiplication.
2yyy=2
y
3
2yyy=2
y
3
.
The repeated factor is
y
y
. The exponent 3 records the fact that the factor
y
y
appears 3 times in the multiplication.
(a+b)(a+b)(a−b)(a−b)(a−b)=
(a+b)
2
(a−b)
3
(a+b)(a+b)(a−b)(a−b)(a−b)=
(a+b)
2
(a−b)
3
.
The repeated factors are
(a+b)
(a+b)
and
(a−b)
(a−b)
,
(a+b)
(a+b)
appearing 2 times and
(a−b)
(a−b)
appearing 3 times.
Write each of the following using exponents.
(3b)(3b)(5c)(5c)(5c)(5c)
(3b)(3b)(5c)(5c)(5c)(5c)
(3b)
2
(5c)
4
(3b)
2
(5c)
4
2⋅2⋅7⋅7⋅7⋅(a−4)(a−4)
2⋅2⋅7⋅7⋅7⋅(a−4)(a−4)
2
2
⋅
7
3
(a−4)
2
2
2
⋅
7
3
(a−4)
2
It is extremely important to realize and remember that an exponent applies only to the factor to which it is directly connected.
8
x
3
8
x
3
means
8⋅xxx
8⋅xxx
and not
8x8x8x
8x8x8x
. The exponent 3 applies only to the factor
x
x
since it is only to the factor
x
x
that the 3 is connected.
(8x)
3
(8x)
3
means
(8x)(8x)(8x)
(8x)(8x)(8x)
since the parentheses indicate that the exponent 3 is directly connected to the factor
8x
8x
. Remember that the grouping symbols
(
)
(
)
indicate that the quantities inside are to be considered as one single number.
34
(a+1)
2
34
(a+1)
2
means
34⋅(a+1)(a+1)
34⋅(a+1)(a+1)
since the exponent 2 applies only to the factor
(a+1)
(a+1)
.
Write each of the following without exponents.
(4a)(4a)(4a)
(4a)(4a)(4a)
Select a number to show that
(2x)
2
(2x)
2
is not always equal to
2
x
2
2
x
2
.
Suppose we choose
x
x
to be 5. Consider both
(2x)
2
(2x)
2
and
2
x
2
2
x
2
.
(2x)
2
2
x
2
(2⋅5)
2
2⋅
5
2
(10)
2
2⋅25
100
≠
50
(2x)
2
2
x
2
(2⋅5)
2
2⋅
5
2
(10)
2
2⋅25
100
≠
50
(1)
Notice that
(2x)
2
=2
x
2
(2x)
2
=2
x
2
only when
x=0
x=0
.
Select a number to show that
(5x)
2
(5x)
2
is not always equal to
5
x
2
5
x
2
.
Select
x=3
x=3
. Then
(5⋅3)
2
=
(15)
2
=225
(5⋅3)
2
=
(15)
2
=225
, but
5⋅
3
2
=5⋅9=45
5⋅
3
2
=5⋅9=45
.
225≠45
225≠45
.
In
x
n
x
n
,
x
x
is the base
n
n
is the exponent
The number represented by
x
n
x
n
is called a power.
The term
x
n
x
n
is read as "
x
x
to the
n
n
th power," or more simply as "
x
x
to the
n
n
th."
The symbol
x
2
x
2
is often read as "
x
x
squared," and
x
3
x
3
is often read as "
x
x
cubed." A natural question is "Why are geometric terms appearing in the exponent expression?" The answer for
x
3
x
3
is this:
x
3
x
3
means
x⋅x⋅x
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
x
then the volume of the cube is
x⋅x⋅x
x⋅x⋅x
, which, of course, is described by
x
3
x
3
. (Can you think of why
x
2
x
2
is read as
x
x
squared?)
Cube with
length
=x
=x
width
=x
=x
depth
=x
=x
Volume
=xxx=
x
3
=xxx=
x
3
In Section (Reference) 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.
- Perform all operations inside grouping symbols beginning with the innermost set.
- Perform all exponential operations as you come to them, moving left-to-right.
- Perform all multiplications and divisions as you come to them, moving left-to-right.
- Perform all additions and subtractions as you come to them, moving left-to-right.
Use the order of operations to simplify each of the following.
2
2
+5=4+5=9
2
2
+5=4+5=9
5
2
+
3
2
+10=25+9+10=44
5
2
+
3
2
+10=25+9+10=44
2
2
+(5)(8)−1
=4+(5)(8)−1
=4+40−1
=43
2
2
+(5)(8)−1
=4+(5)(8)−1
=4+40−1
=43
7⋅6−
4
2
+
1
5
=7⋅6−16+1
=42−16+1
=27
7⋅6−
4
2
+
1
5
=7⋅6−16+1
=42−16+1
=27
(2+3)
3
+
7
2
−3
(4+1)
2
=
(5)
3
+
7
2
−3
(5)
2
=125+49−3(25)
=125+49−75
=99
(2+3)
3
+
7
2
−3
(4+1)
2
=
(5)
3
+
7
2
−3
(5)
2
=125+49−3(25)
=125+49−75
=99
[4
(6+2)
3
]
2
=
[4
(8)
3
]
2
=
[4(512)]
2
=
[2048]
2
=4,194,304
[4
(6+2)
3
]
2
=
[4
(8)
3
]
2
=
[4(512)]
2
=
[2048]
2
=4,194,304
6(
3
2
+
2
2
)+
4
2
=6(9+4)+
4
2
=6(13)+
4
2
=6(13)+16
=78+16
=94
6(
3
2
+
2
2
)+
4
2
=6(9+4)+
4
2
=6(13)+
4
2
=6(13)+16
=78+16
=94
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
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
Use the order of operations to simplify the following.
2
3
+
3
3
−8⋅4
2
3
+
3
3
−8⋅4
1
4
+
(
2
2
+4)
2
÷
2
3
1
4
+
(
2
2
+4)
2
÷
2
3
[6(10−
2
3
)]
2
−
10
2
−
6
2
[6(10−
2
3
)]
2
−
10
2
−
6
2
5
2
+
6
2
−10
1+
4
2
+
0
4
−
0
5
7
2
−6⋅
2
3
5
2
+
6
2
−10
1+
4
2
+
0
4
−
0
5
7
2
−6⋅
2
3
For the following problems, write each of the quantities using exponential notation.
3 squared times
y
y
to the fifth
a
a
cubed minus
(b+7)
(b+7)
squared
a
3
−
(
b+7
)
2
a
3
−
(
b+7
)
2
(21−x)
(21−x)
cubed plus
(x+5)
(x+5)
to the seventh
2⋅3⋅3⋅3⋅3xxyyyyy
2⋅3⋅3⋅3⋅3xxyyyyy
2(
3
4
)
x
2
y
5
2(
3
4
)
x
2
y
5
2⋅2⋅5⋅6⋅6⋅6xyyzzzwwww
2⋅2⋅5⋅6⋅6⋅6xyyzzzwwww
7xx(a+8)(a+8)
7xx(a+8)(a+8)
7
x
2
(
a+8
)
2
7
x
2
(
a+8
)
2
10xyy(c+5)(c+5)(c+5)
10xyy(c+5)(c+5)(c+5)
(
4x
)
5
or
4
5
x
5
(
4x
)
5
or
4
5
x
5
(9a)(9a)(9a)(9a)
(9a)(9a)(9a)(9a)
(−7)(−7)(−7)aabbba(−7)baab
(−7)(−7)(−7)aabbba(−7)baab
(
−7
)
4
a
5
b
5
(
−7
)
4
a
5
b
5
(a−10)(a−10)(a+10)
(a−10)(a−10)(a+10)
(z+w)(z+w)(z+w)(z−w)(z−w)
(z+w)(z+w)(z+w)(z−w)(z−w)
(
z+w
)
3
(
z−w
)
2
(
z+w
)
3
(
z−w
)
2
(2y)(2y)2y2y
(2y)(2y)2y2y
3xyxxy−(x+1)(x+1)(x+1)
3xyxxy−(x+1)(x+1)(x+1)
3
x
3
y
2
−
(
x+1
)
3
3
x
3
y
2
−
(
x+1
)
3
For the following problems, expand the quantities so that no exponents appear.
8 · x · x · x · y · y
8 · x · x · x · y · y
(18
x
2
y
4
)
2
(18
x
2
y
4
)
2
(9
a
3
b
2
)
3
(9
a
3
b
2
)
3
(
9aaabb
)(
9aaabb
)(
9aaabb
) or 9 · 9 · 9aaaaaaaaabbbbbb
(
9aaabb
)(
9aaabb
)(
9aaabb
) or 9 · 9 · 9aaaaaaaaabbbbbb
5
x
2
(2
y
3
)
3
5
x
2
(2
y
3
)
3
10
a
3
b
2
(3c)
2
10
a
3
b
2
(3c)
2
10aaabb(
3c
)(
3c
) or 10 · 3 · 3aaabbcc
10aaabb(
3c
)(
3c
) or 10 · 3 · 3aaabbcc
(a+10)
2
(
a
2
+10)
2
(a+10)
2
(
a
2
+10)
2
(
x
2
−
y
2
)(
x
2
+
y
2
)
(
x
2
−
y
2
)(
x
2
+
y
2
)
(
xx−yy
)(
xx+yy
)
(
xx−yy
)(
xx+yy
)
For the following problems, select a number (or numbers) to show that
(5x)
2
(5x)
2
is not generally equal to
5
x
2
5
x
2
.
(7x)
2
(7x)
2
is not generally equal to
7
x
2
7
x
2
.
Select
x=2.
x=2.
Then,
196≠28.
196≠28.
(a+b)
2
(a+b)
2
is not generally equal to
a
2
+
b
2
a
2
+
b
2
.
For what real number is
(6a)
2
(6a)
2
equal to
6
a
2
6
a
2
?
For what real numbers,
a
a
and
b
b
, is
(a+b)
2
(a+b)
2
equal to
a
2
+
b
2
a
2
+
b
2
?
Use the order of operations to simplify the quantities for the following problems.
8
2
+3+5(2+7)
8
2
+3+5(2+7)
3
4
+
2
4
(1+5)
3
3
4
+
2
4
(1+5)
3
(
6
2
−
4
2
)÷5
(
6
2
−
4
2
)÷5
2
2
(10−
2
3
)
2
2
(10−
2
3
)
(
3
4
−
4
3
)÷17
(
3
4
−
4
3
)÷17
(4+3)
2
+1÷(2⋅5)
(4+3)
2
+1÷(2⋅5)
(
2
4
+
2
5
−
2
3
⋅5)
2
÷
4
2
(
2
4
+
2
5
−
2
3
⋅5)
2
÷
4
2
1
6
+
0
8
+
5
2
(2+8)
3
1
6
+
0
8
+
5
2
(2+8)
3
(7)(16)−
9
2
+4(
1
1
+
3
2
)
(7)(16)−
9
2
+4(
1
1
+
3
2
)
(1+6)
2
+2
19
(1+6)
2
+2
19
6
2
−1
5
+
4
3
+(2)(3)
10
6
2
−1
5
+
4
3
+(2)(3)
10
5[
8
2
−9(6)]
2
5
−7
+
7
2
−
4
2
2
4
−5
5[
8
2
−9(6)]
2
5
−7
+
7
2
−
4
2
2
4
−5
(2+1)
3
+
2
3
+
1
3
6
2
−
15
2
−
[2(5)]
2
5⋅
5
2
(2+1)
3
+
2
3
+
1
3
6
2
−
15
2
−
[2(5)]
2
5⋅
5
2
6
3
−2⋅
10
2
2
2
+
18(
2
3
+
7
2
)
2(19)−
3
3
6
3
−2⋅
10
2
2
2
+
18(
2
3
+
7
2
)
2(19)−
3
3
1070
11
or 97.
27
¯
1070
11
or 97.
27
¯
((Reference)) Use algebraic notation to write the statement "a number divided by eight, plus five, is equal to ten."
((Reference)) Draw a number line that extends from
−5
−5
to 5 and place points at all real numbers that are strictly greater than
−3
−3
but less than or equal to 2.
((Reference)) Use the commutative property of multiplication to write a number equal to the number
yx
yx
.
((Reference)) Use the distributive property to expand
3(x+6)
3(x+6)
.
(What does "Endorsed by" mean?)
"Reviewer's Comments: 'I recommend this book for courses in elementary algebra. The chapters are fairly clear and comprehensible, making them quite readable. The authors do a particularly nice job […]"