Exponents: Exponentiation and Example
Exponents and Powers
Very large numbers and very small numbers are difficult to read, understand, and compare.
To make this easier, we use exponents by converting many of the large numbers and small numbers into a shorter form.
For example: 10,000,000,000,000 can be written as (10)13.
Here, 10 is called the base and 13 is called the exponent.
For any non-zero integer a, a m
1
, where m is a positive integer. am a–m is called the multiplicative inverse of am and vice-versa.
Decimal numbers can be written in expanded form using exponents.
For example: The number, 32845.912 can be written in an expanded form as follows.
32845.912
1
1
1
1 2 2 3
10
10
10
4
3
2
1
1
2
3 10 2 10 8 10 4 10 5 1 9 10 110 2 10 3
3 104 2 103 8 102 4 101 5 1 9
Laws of exponents
am an = am + n
am
a mn an a 0
(am)n = amn
am bm = (ab)m
am a
bm b
a
b
m
a0 = 1
m
b
a
m
(a 0)
Example: Simplify
Solution:
4
2
54 33 1
21 .
23 15
5
2
4
2
54 33 1
21
3
2 15
5
2
2
2
2
54 33
23 3 5
1 1
2
5 2
24 54 33
231 3 5
1 2
a m a m
1
m
m and a m b a
b
a m n a mn and a m a n a m n
24 54 33
24 3 53
2
44
5
4 3
31
3
am mn
n a
b
20 51 32
1 5 9
a 0 1
45
an = 1 n = 0 for any a 0, except at a = 1 or a = –1
Note: (1)n = 1 for any n
(–1)p = 1 for any even integer p
Very smaller numbers can be expressed in a simpler way using negative exponents.
Example: Write the number, 0.00000003812, in standard form.
Solution:
We have,
3812
3812 3.812 103
0.00000003812
11
3.812 10311 3.812 108
11
Very large numbers and very small numbers are difficult to read, understand, and compare.
To make this easier, we use exponents by converting many of the large numbers and small numbers into a shorter form.
For example: 10,000,000,000,000 can be written as (10)13.
Here, 10 is called the base and 13 is called the exponent.
For any non-zero integer a, a m
1
, where m is a positive integer. am a–m is called the multiplicative inverse of am and vice-versa.
Decimal numbers can be written in expanded form using exponents.
For example: The number, 32845.912 can be written in an expanded form as follows.
32845.912
1
1
1
1 2 2 3
10
10
10
4
3
2
1
1
2
3 10 2 10 8 10 4 10 5 1 9 10 110 2 10 3
3 104 2 103 8 102 4 101 5 1 9
Laws of exponents
am an = am + n
am
a mn an a 0
(am)n = amn
am bm = (ab)m
am a
bm b
a
b
m
a0 = 1
m
b
a
m
(a 0)
Example: Simplify
Solution:
4
2
54 33 1
21 .
23 15
5
2
4
2
54 33 1
21
3
2 15
5
2
2
2
2
54 33
23 3 5
1 1
2
5 2
24 54 33
231 3 5
1 2
a m a m
1
m
m and a m b a
b
a m n a mn and a m a n a m n
24 54 33
24 3 53
2
44
5
4 3
31
3
am mn
n a
b
20 51 32
1 5 9
a 0 1
45
an = 1 n = 0 for any a 0, except at a = 1 or a = –1
Note: (1)n = 1 for any n
(–1)p = 1 for any even integer p
Very smaller numbers can be expressed in a simpler way using negative exponents.
Example: Write the number, 0.00000003812, in standard form.
Solution:
We have,
3812
3812 3.812 103
0.00000003812
11
3.812 10311 3.812 108
11