1. How do we handle fractional exponents?
Do Now:
28 =
256
Fill in the appropriate
information
27 =
2? =
128
26 = 64
2? =
25 = 32
24 =
2? = 16
23 = 8
22 = 4
21 = 2
2-1 =
1/2
2-2 =
2? =
1/4
2-3 =
2? =
1/8

2. How do we handle fractional exponents?
Do Now:
Simplify/Rationalize:

3. Properties of Exponents
Product of Powers Property
am • an = am+n
Power of Power Property
(am)n = am•n
Power of Product Property
(ab)m = ambm
Negative Power Property
a-n = 1/an, a 0
Zero Power Property
a0 = 1
Quotients of Powers Property
Power of Quotient Property

4. Indices, Exponents, and New Power Rules
Product of Powers Property
am • an = am+n
example:
82 • 83 = = 85
example:
x3 • x6 = x3 + 6 = x9
Power of Product Property
(ab)m = am • bm
example:
(2 • 8)2 = 22 • 82
example:
(xy)5 = x5 • y5
Power of Quotient Property
example:

5. beware! -23/2 is not the same as (-2)3/2
Types of exponents
Positive Integer Exponent
an = a • a • a • • • • a
n factors
Zero Exponent
a0 = 1
Negative Exponent -n
Rational Exponent 1/n
Rational Expo. m/n
- m/n
Negative
Rational Exponent
beware! /2 is not the same as (-2)3/2

6. Roots, Radicals & Rational Exponents
radical sign
radicand
index
of a number is one of the two
equal factors whose product is
that number
Square Root
has an index of 2
can be written exponentially as
Every positive real number
has two square roots
The principal square root of
a positive number k is its
positive square root,
If k < 0, is an imaginary
number

7. Roots, Radicals & Rational Exponents
radical sign
radicand
index
Cube Root
of a number is one of the three
equal factors whose product is
that number
has an index of 3
can be written exponentially as
k1/3
principal cube roots

8. Roots, Radicals & Rational Exponents
radical sign
radicand
index
of a number is one of n
equal factors whose product is
that number
nth Root
has an index where n is any
counting number
can be written exponentially as
k1/n
principal odd roots
principal even roots

9. Indices and Rational Exponents
square root -
k1/2 • k1/2 = k 1/2 + 1/2 = k1 = k
21/2 • 21/2 = 2 1/2 + 1/2 = 21 = 2
cube root -
k1/3 • k1/3 • k1/3 = k 1/3 + 1/3 + 1/3 = k1 = k
21/3 • 21/3 • 21/3 = 2 1/3 + 1/3 + 1/3 = 21 = 2
nth root -
= k1/n
k1/n • k1/n • k1/n = k 1/n + 1/n + 1/n = k1 = k
n times
n times
81/3 • 81/3 • 81/3 = 8 1/3 + 1/3 + 1/3 = 81 = 8

10. Fractional Exponents
Radicals
Fractional Exponents
(ab)m = am • bm

11. Simplifying = x10 - 4 = x5 - 2 = x6 = x3 positive integer exponent
rational exponent
multiplication law
simplify &
power law
division law
= x10 - 4
= x5 - 2
= x6
= x3

12. Simplifying – Fractional Exponents
A rational expression that contains a fractional exponent in the denominator must also be rationalized. When you simplify an expression, be sure your answer meets all of the given conditions.
Conditions for a Simplified Expression
It has no negative exponents.
It has no fractional exponents in the denominator.
It is not a complex fraction.
The index of any remaining radical is as small as possible.

13. Model Problems
Rewrite using radicals:
Rewrite using rational exponents:
Evaluate:

14. Evaluate a0 + a1/3 + a -2 when a = 8
Evaluating
Evaluate a0 + a1/3 + a -2 when a = 8 /
replace a with 8
1 + 81/
x0 = 1
x1/3 =
x–n = 1/xn
8–2 = 1/82 = 1/64
/64
3 1/64
combine like terms
If m = 8, find the value of (8m0)2/3
(8 • 80)2/3
replace m with 8
(8 • 1)2/3
x0 = 1
(8)2/3
= 4

15. Simplifying – Fractional Exponents