Warm up
• 1. Change into Scientific Notation
• 3,670,900,000
• Use 3 significant figures
• 2. Compute: (6 x 102) x (4 x 10-5)
• 3. Compute: (2 x 10 7) / (8 x 103)
Lesson11-1 Rational Exponents
Objective: To use the properties of exponents
To evaluate and simplify expressions containing rational
exponents
To solve equations containing rational exponents
Nth Roots
a is the square
root of b if a2 = b
a is the cube root
of b if a3 = b
Therefore in
general we can
say:
a is an nth root of
b if an = b
Exponents
n
n=1: b =b
n>1: b =b  b  b  ...  b
n
0
n=0: b =1
1
If b  0, then b = n .
b
-n
Properties of Exponents
m and n are positive integers
a and b are real numbers
Property
Definition
Example
Product
aman=am+n
(am)n=amn
(163)(167)=1610
(93)2=96
Power of a power
Power of a quotient (a/b)m=am/bm
(b≠0)
(3/4)5=35/45
Power of a product
(ab)m=ambm
(5x)3=53x3
quotient
am/an=am-n
156/152=154
(a≠0)
Example
Evaluate:
3 3

6
3
4
7
1
3
  
4
4 7 6
3
4
3
 3
5
Example
Simplify:
s t 
4 7 3
x2 y5
x 
2 4


12 21
s t
x2 y5

8
x
y5
x6
Nth Root
• If b > 0 then a and –a are square roots of b
• Ex: 4 = √16 and -4 = √16
• If b < 0 then there are not real number square
roots.
• Also b1/n is an nth root of b.
• 1441/2 is another way of showing √144
• ( =12)
Principal nth Root
• If n is even and b is positive, there are two
numbers that are nth roots of b.
• Ex: 361/2 = 6 and -6 so if n is even (in this case 2) and
b is positive (in this case 36) then we always choose
the positive number to be the principal root. (6).
The principal nth root of a real number b, n > 2 an integer,
symbolized by n
means an = b
b a
if n is even, a ≥ 0 and b ≥ 0
if n is odd, a, b can be any real number
index
n
radical
b
radicand
Examples
Find the principal root:
1.
5
1
2. 811/2
Evaluate
3. (-8)1/3
4. -(
1 1/4
)
16
Rational Exponents
For any nonzero number b, and any integers m
and n with n>1, and m and n have no common
factors
m
n
b  b 
n
m
 b
n
m
.
(Except where b1/n is not a real number)
Properties of Powers an = b and
Roots a = b1/n for Integer n>0
• The even root of a negative number is not a
real number.
• Ex: (-9)1/2 is undefined in the real number system
Example
Evaluate:
4
5
4
5 5
243   3
27
27
5
3
2
3
5
3


 27  27
3  81
4
2
3
3
3
 27  27
Example
Express using rational exponents.
1
4 12 16 4
2 s
t


3 4
2s t
Express using radicands.
3
4
1
2
5x y  5  x y
3
1
2 4


4
3
5 x y
2
Example
Simplify.
7 5 24
r st

3
2 12
r st
rs
Avoid negative values that
would result in an imaginary
number!
Rational Exponents
• bm/n = (b1/n)m = (bm)1/n
• b must be positive when n is even.
• Then all the rules of exponents apply when the
exponents are rational numbers.
• Ex: x⅓ • x ½ =
•
x ⅓+ ½ =
•
x5/6
• Ex: (y ⅓)2 = y2/3
Radicals
•
bm/n = (bm)1/n =
m
and
bm/n
=
(b1/n)m
=(
Ex: 82/3 = (82)1/3 =
= (81/3)2= (
)
m
2
)2
Properties of Radicals
•
m
=(
•
m
)
2
=
•
=
=
•
n
•
n
=
3
= a if n is odd
= │a │if n is even
)2 = 4
=(
2
=6
=
= -2
2
(2) = │-2│= 2
Simplifying Radicals
• Radicals are considered in simplest form when:
• The denominator is free of radicals
m
•
has no common factors between m and n
m
•
has m < n
Practice
• Simplify
• (a1/2b-2)-2 =
•
=
Solving equations
4
5
Solve 616  x  9.
625  x
4
5
isolate the variable
5
4
625  x
5
4 4
5 
x
3125  x
Calculator: 625^(5÷4)
in class assignment
• p700
• #4-18 all
Sources
• http://www.onlinemathtutor.org/help/mathcartoons/mr-atwadders-math-tests/
• images.pcmac.org/SiSFiles/Schools/AL/.../BJHi
gh/.../11-1Exponents.ppt