Download Exponents and Radicals Notes

Exponents and Radicals
A product of identical numbers is usually written in exponential notation. For example,
5 . 5 . 5 = 53 . In general,
we have the following definition.
!Exponential Notation: If a is any real number and n is a positive integer, then the
n
nth ower of a is
The number a is called the base and the number n is called the ex
'----------.
a .
Example 1: Exponential
notation
\
'-
0
\
. '-O-
\
-
d-
d
\
-
Notice a pattern from example 1 part b? What did you do with the exponents?
Zeros and Negative EXp'onents:
anda
-n
If
a;j:.
0 is an real number and
n
is a ositive integer, then aO =
1
=~.
a
Example 2: Zero and negative exponents.
a)
.
\
--
-d.-
1
.:
Laws of Exponents:
Description
Law
~ti)Q
l.
aman
-
a n+ n
To multiply ~owers
of the same
number, add t e exponents
To divide two powers of the same
number, subtract the exponents.
m
2.
a
-=a
an
3.
(am
4.
(aby ==a nbn
To raise a product to a power, raise
each factor to the power.
J = ~:
6(:r =(!J
To raise a quotient to a power, raise
both numerator and denominator to
the power.
m=n
t ==a m·n
To raise a power to a new power,
multiply the exponents.
5. (~
7.
a -n
b-m
To raise a fraction to a negative
power, invert the fraction and change
the sign of the exponent.
bm
-an
-
To move a number raised to a power
form numerator to denominator or
from denominator to numerator,
change the sign of the exponent.
Example 3: Simplifying expressions with exponents
a) (2a3b2)(3ab4)3
= (J~O:~ <> ') l3
3
- (,)03'0"
Sb)
6st-4
2s-2t2
~
ClocI)')
&~\5~)
L{ CL~ \01 '-I
d
V ~ ..~
C} t ~t:-J
LOUJJ
3
Rational Exponents:
(\3
~
I
_\
~
~
WOJV\/-\-
GtJuld ~
Yl~ ~ ~
efinition of rational exponents:'--__ --------~......,
or any rational exponent mln in lowest terms, wherem and n are integers and n > 0, we define
a'% =
(~t
or equivalently
a%
= ~.
[f n is even, then we re uire a ~ 0 .
Example 4: Rational Exponents
li
a) 4
\
\
Rationalizing the Denominator
It is often useful to eliminate the radical in a denominator by multiplying both the numerator and
denominator by an appropriate expression.
2
a) -
J3
•
1
b)
ifI2~
3 X
n~
•
~
-
~
c)
V:2
~
~
~o1
~
~
...,
--
-:
~ ~·M~~
L~
K
)(d~ ~
\
•
X$
~o!l
Zj&1
Sl
~
Q::>
~