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Lesson 5.1
Exponents
Definition of Power:
If b and n are counting numbers, except that b and n are not both zero,
then exponentiation assigns to b and n a unique counting number bn,
called a power.
b is called the base.
bn
n is called the exponent.
How do we compute exponents?
The exponent counts the number of times that 1 is multiplied by the base.
OR
The exponent
joined to 1.
n counts the number of times that the expansion · b is
Power
Form
Number of
Expansions
34
four
1·3·3·3·3
81
33
three
1·3·3·3
27
32
two
1·3·3
9
31
one
1·3
3
30
none
1
1
In general,
Operator
Model
bn = 1 · b · b · b · … · b
n times
Basic
Numeral
Here are several more examples:
33 = 1 · 3 · 3 · 3 = 27
32 = 1 · 3 · 3 = 9
31 = 1 · 3 = 3
30 = 1
23 = 1 · 2 · 2 · 2 = 8
22 = 1 · 2 · 2 = 4
21 = 1 · 2 = 2
20 = 1
13 = 1 · 1 · 1 · 1 = 1
12 = 1 · 1 · 1 = 1
11 = 1 · 1 = 1
10 = 1
03 = 1 · 0 · 0 · 0 = 0
02 = 1 · 0 · 0 = 0
01 = 1 · 0 = 0
??
We have the following general rules:
For all b, b1 = b.
What about
00
?
For all b  0,
It is not defined.
b0 = 1.
For all n  0,
0n = 0.
Evaluate each power:
62
By definition, 62 = 1 · 6 · 6 = 36
When the exponent is greater than 1, this can be shortened
by dropping the 1 at the front and writing: 62 = 6 · 6 = 36.
24
We may compute this as 24 = 2 · 2 · 2 · 2 = 16
50
Since the exponent is 0, we have to use the definition.
50 is 1 multiplied by 5 no times. That is, 50 = 1.
We may use exponents to count the number of factors:
7 · 7 · 7 · 7 may be written as
74.
b · b · b may be written as
b3.
5 · 3 · 3 · 5 · 3 · 3 may be written as 34 ·
5 2.
Caution: You may have been told that 23 is 2 times itself 3 times.
This is NOT TRUE !!
We may write
23 = 2 · 2 · 2,
but there are only 2 products here, not 3.
Remember: For 23, the exponent 3 counts the number of times
that
1 is multiplied by 2.
23 = 1 · 2 · 2 ·
2