Transcript Ch 6.1
Roots of Real
Numbers and Radical
Expressions
Definition of
th
n
Root
For any real numbers a and b
and any positive integers n,
if b^n = a,
then b is the nth root of a.
** For a square root the value of n is 2.
Notation
radical
index
4
81
radicand
Note: An index of 2 is understood but not
written in a square root sign.
4
Simplify
81
To simplify means to find x
in the equation:
4
x = 81
Solution:
4
81=
3
Principal Root
The nonnegative root of a number
64
Principal square root
64
Opposite of principal
square root
64
Both square roots
Summary of Roots
a
The n th root of a
n a>0 a<0 a=0
n
even
odd
one + root
one - root
one + root
no - roots
no real
roots
no + roots
one - root
one real
root, 0
Rational
Exponents
In other words,
exponents that are
fractions.
Definition of
a
1
n
For any real number a
and any integer n > 1,
1
n
a a
except when a < 0 and n
is even
n
Examples:
1
36 6
1. 36
2
1
2. 64
3
3
64 4
Examples:
1
1 2
2
3. 49
49
1
1 1
49 7
1
1
4. 83
8
3
1
3
82
Definition of Rational Exponents
For any nonzero number b and
any integers m and n with n > 1,
a
except when b < 0 and n is even
m
n
a a
n
m
n
m
NOTE: There are 3 different
ways to write a rational
exponent
4
27 27
3
3
4
3
4
27
Examples:
36
6 216
1. 36
2. 27 27 3 81
81
3 27
3. 81
3
3
2
4
3
3
3
4
4
3
4
4
3
3