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Roots of Real
Numbers and Radical
Expressions
Definition of
th
n
Root
For any real numbers a and b
and any positive integers n,
n
if a = b,
then a is the nth root of b.
** 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
b
The n th root of b
n b >0 b <0 b =0
n
even
odd
no real
one (+) root
one real
roots
one (-) root
root, 0
one (+) root no (+) roots
no (-) roots one (-) root
Examples


13
x

1.  169 x
4
2. -


8
x
3


4

2 2
 13x
  8 x  3 
2 2
   8 x  3
2
2
Examples
3.
4.
3

125 x
3
6
3
m n 
3
3
5 x 
3
2
3
 5x
2
 mn  mn
3
When to use absolute value signs
If the index (n) of the radical is
even, the power under the radical
sign is even, and the resulting
power is odd, then we must use an
absolute value sign.
Even – Even – Odd ….
ABSOLUTE VALUE SIGNS!
Examples
Even
Odd
Even
1.
4
 an   an
4
Even
2.
6
 xy 
6
2
Even
xy
Odd
2
Even
Odd
Even
2
3.
x
6
 x
3
Odd
Even
4.
6
3  y 
2 18
 3 - y
3
2
Even
 3 - y

3
2
Classwork:
1) 3
2)
64
(2)
3)
5
4)
4
5) 3
2
6) 4
 243
7)
 4096
8)
x
w
3
4
2 4
36a b
(4 x  3 y )
2