Section 1.2 Exponents & Radicals
Objectives:•To review exponent rules•To review radicals•To review rational exponents
Integer Exponents
factorsn an = a · a · · · · · a
The number a is called the base and n is called the exponent.
Ex 1. Simplify
512
4
4
(a)
(b) 3
(c) 3
1 1 1 1 1 12 2 2 2 2 32
3 3 3 3 81
3 3 3 3 81
Zero & Negative Exponents
If a ≠ 0 is any real number and n is a positive integer, then
a0 = 1 and a–n = 1na
Ex 2. Simplify
047
1
3
(a)
(b)
(c) 2
x
1
1
x
3
1 1
82
Laws of ExponentsLaw Example
1)
2)
3)
4)
5)
m na a 2 33 3
m
n
a
a
7
4
5
5
nma 352
nab 6xy
na
b
25
7
Class Work
Simplify1. 2.
3. 4.
5.
4 7x x 4 7y y
9
5
c
c
33 2 42 3s t st
34 3 2
2 3
a b b
ab a
Laws for Negative Exponents
Law Example
6.
na
b
nb
a
23
4
24
3
7.
n
m
a
b
m
n
b
a
2
5
3
4
5
2
4
3
Ex 3 Simplify
a) b)
4
2 2
6
2
st
s t
2
33
y
z
Class Work
6.
7.
32 3
2 3 4
xy z
x y z
22
23
23
a bab c
c
Radicals
The symbol √ means: “the positive square root of.”
Thus,
2means and 0a b b a b
nth Roots
If 24 = 16, then it follows that .
4 16 2
If n is any positive integer, then the principal nth root of a is defined as follows:
If n is even, we must have a ≥ 0 and b ≥ 0.
means nn a b b a
Ex 4. Simplify
a)
b)
c)
d)
4 81
5 32
3 8
4 16
3
2
2
not defined
Properties of nth Roots
Property Example1.
2.
3.
n n nab a b 3 8 27
n
nn
a a
b b 4
16
81
3 38 27 2 3 6
4
4
16 2
381
m n mna a3 729 6 729 3
Property Example
4.
5.
n na a if n is odd 33 5 5
n na a if n is even 44 5 5 5
Ex 5. Simplify
3 4)a x
8 44) 81b x y
) 32 200c
3x x
23x y
4 2 10 2 14 2
Class Work
8.
9.
10.
4 2 124 x y z 3 24xz y
3 32 4a b a b 3 36 2 2 2a b a b
75 48 5 3 4 3 9 3
Rational Exponents
For example,
and,
For example,
1/ n na a
31/ 3x x
/ /( ) or equivalentlym n m m n mnna a a a
23 32 / 3 2x x or x
Ex 6. Simplify
1/ 2
2 / 3
1/ 3
(a) 4
(b) 8
(c) 125
Class Work
11.
12.
13.
14.
1/ 3 7 / 3a a
2 / 5 7 / 5
3 / 5
a a
a
3 / 23 42a b
33 / 4 4
1/ 3 1/ 2
2x y
y x
8 / 3a
9 / 5
6 53 / 5
aa
a3 2 9 2 62 a b
9 4
4 1 2 11 4 388
xy x x y
y
HW #2 p21 1-7odd, 9,12,15,17,23-43odd,
45-65eoo,eoo – every other odd