Download Examples

5.5 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
n
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
The nth root of b
n
b>0
b<0
even
odd
one + root
one - root
no real
roots
n
b
b=0
one real
root,
0
one + root no + roots
no - roots one - root
Examples
1.  169x
2. -
4
8x- 3
4
Examples
3.
4.
3
3
125x
6
m n
3 3
Taking nth roots of variable
expressions
Using 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.
Examples
Odd
Even
1.
4
Even
2.
6
an  an
Even
4
xy 
2
6
Odd
Even
= xy
2
Odd
Even
3.
2
x
Even
4.
6
6
 x
Even
3
Odd
3  y  = 3 - y 
2
2
= 3- y
18
3
Even
2
3
Product Property of
Radicals
For any numbers a and
b where a 0and b 0,
ab  a  b
Product Property of
Radicals Examples
72 
36 2  36 2
6 2
48  16 3  16 3
4 3
What to do when the index will not
divide evenly into the radical????
• Smartboard Examples..\..\Algebra II
Honors 2007-2008\Chapter 5\5.5
Simplifying Radicals\Simplifying
Radicals.notebook
Examples:
1.
30a  a  30
34
34
 a
17
2.
30
54x y z  9x y z  6yz
4 4 6
4 5 7
 3x y z
2
2
3
6 yz
Examples:
3.
3
54a b  27a b  2b
3 7
3
3 7
3
 3ab  2b
2
4.
3
60xy  4 y  15xy
3
2
 2 y 15xy
Quotient Property of
Radicals
For any numbers a and
b where a 0and b 0,
a

b
a
b
Examples:
1.
7

16
32

2.
25
7
7

4
16
32
25

32 4 2

5
5
Examples:
3.
4.
48
3

45

4
48

3
16  4
45
45 3 5

2
2
4

Rationalizing the
denominator
Rationalizing the denominator means
to remove any radicals from the
denominator.
Ex: Simplify
5

3
5
3

3
3

5 3
9

5 3

3
15
3
Simplest Radical Form
•No perfect nth power factors
other than 1.
•No fractions in the radicand.
•No radicals in the denominator.
Examples:
1.
5

4
20 8
5
5

2
4
8
2.
 10  10 4  10 2
2
2 2
 20
Examples:
5
2
5 2
5
2

3.



4
2 2 2 2 4
2 2
5 2
5 4 5 7x 4 35x
4. 4



2
7x
7x
7x
49x
4 35x

7x
Adding radicals
We can only combine terms with radicals
if we have like radicals
6 7 5 7  3 7
 6 5 3 7  8 7
Reverse of the Distributive Property
Examples:
1. 2 3 + 5 + 7 3 - 2
= 2 3 + 7 3 + 5- 2
= 9 3+3
Examples:
2. 5 6  3 24  150
= 5 6  3 4 6  25 6
= 5 6 6 6  5 6
=4 6
Multiplying radicals Distributive Property
 2  4 3
3

3 2  3 4 3

6  12
Multiplying radicals - FOIL

3 5
F


24 3
O
3 2  3 4 3
I

L
 5 2  5 4 3

6  12 10  4 15
Examples:



1. 2 3  4 5 3  6 5
O
F
 2 3 3  2 3 6 5
L
I
4 5  3  4 5  6 5
 6  12 15  4 15  120
 16 15  126
Examples:

5 4  2 7
= 5 2 2 75 2 2 7
O
F
2. 5 4  2 7
 1010 10 2 7
I
L
2 7 10 2 7  2 7
 100 20 7  20 7  4 49
 100 4 7  72
Conjugates
Binomials of the form
a b  c d anda b  c d
where a, b, c, d are rational
numbers.
Ex:
5  6  Conjugate:
56
3  2 2  Conjugate: 3  2 2
What is conjugate of
2 7  3?
Answer: 2 7  3
The product of conjugates is a
rational number. Therefore, we can
rationalize denominator of a fraction
by multiplying by its conjugate.
Examples:
32 35

1.
35 35
3 3  5 3  2 3  2 5

2
2
3 5
3 7 3  10 13 7 3


22
3 25
 
Examples:
1 2 5 6  5

2.
6 5 6  5

6 5  12 5 10
6 
2
 
2
5
16 13 5

31