Integer Exponents and Rules of Exponents
Integer Exponents
Definition:
1
x
-1 =
x
and
1
x
-n =
xn
|
Example
3
-1 = 1/3
2
-3
= 1/23 = 1/8
Definition:
x0
= 1
for all nonzero
x. |
Exponent Rule 1
Examples:
-
23 25 = 23 + 5
= 28
-
w2 w3 = w5
-
xy2
x3y3x4y4 = x8y9
Rule 2
Examples:
-
(24)2 = 28
-
(x2)3 = x6
-
42 23 8 = (22)2 23 23
= 24 23 23 = 210
Rule III
Examples:
-
(6)4 = (2)4(3)4
-
(3x2)3 =
33 (x2)3 = 27 x6
-
(-x3)4 = (-1)4 (x3)4
= (1) (x12) = x12
Rule IV
Examples
-
x3
=
x3-2 = x1 = x
x2
-
x4
1
= x4 - 8
=
x-4 =
x8
x4
Application
The population in billions of the world can be modeled by
the equation
P =
A 2t/30
where t is the time in years after 1970. If there were
3 billion people
in 1970, What will the population be in 2060?
Solution:
We first know that if t is 0 then
P is 3 (in billions).
Hence
3 = A 20/30
= A
So that
P =
3 (2t/30)
2060 corresponds to t = 90 so that
P = (3) (290/30)
= (3) (23) = (3)(8) = 24 billion people.
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