*Section 5-2:
Rational
Exponents*
* *
*All rules presented
in
the previous section were defined for integers only. All of the
properties
in the last section can also be extended to include rational exponents
according to the following definitions:*

__Examples__
* *
*1) *

*2) *

* *

*3) *

* *

*4) *

* *

*5) *

* *

*6) *

* *

*7) (100*^{1/2}
- 36^{1/2})^{2}
= (10 - 6)^{2} = 4^{2}
= 16

* *

*8) x*^{1/2}(x^{3/2}
+ 2x^{1/2}) = x^{2} + 2x

* *

*9) *

We can use these rules to
solve
for x when x is the exponent. This method will only work if the
bases
are the same. Check back in section 5-1 for the appropriate rule!!
__Examples__
1) 16^{x} = 2^{5 }
We can write both sides in base two.

2^{4x}
= 2^{5} Now use the fact that the bases are the
same,
the exponents are =

4x = 5 And solve for x!!

x = 5/4 To check it, take the 5/4 root of 16 =
2^{5}!!

2) 27^{1-x} = (1/9)^{3-x}
You need to make both bases the same. How about 3!!

3^{3(1-x)} = 3^{-2(3-x)}
Notice the power on the right side is negative.

3(1 - x) = -2(3 - x) Because the bases are =, the
exponents
must be =

3 - 3x = -6 + 2x Solve for x.

3 = -6 + 5x

9 = 5x

9/5 = x

*The growth
and
decay formula can also be used with rational numbers. Consider
the
following:*
* *
*1) The cost of a
computer
has been increasing at 7% per year. If it costs $1500 now, find
the
cost:*

*
a) 2 years and 6 months from now*

*
b) 3 years and 3 months ago.*

*
Solutions:*

*
a) A*_{o} = 1500, r = .07 and t = 2.5

*
A(2.5) = 1500(1 + .07)*^{2.5}
= 1500(1.07)^{2.5}
=
1776.44

* *

*
b) A*_{o} = 1500, r = .07, and t = -3.25

*
A(-3.25) = 1500(1.07)*^{-3.25}
= 1203.91

*Let's dribble into
the
next section!! *
* *
*I better bounce back
to
the previous section! *
* *