Download 5.4 The Irrational Numbers and the Real Number

Chipola College
MGF 1107
5.4 The Irrational Numbers and the Real Number System_________________________
Irrational number is a real number whose decimal representation is a nonterminating,
nonrepeating decimal number.
Examples/ 6.01020304….. , .353553555…. ,

Determine whether the number is rational or irrational.
1. 4.323223222….
2.
MEMORIZE
Perfect “Squares”
12 = 1
22 = 4
32 = 9
42 = 16
52 = 25
62 = 36
72 = 49
82 = 64
92 = 81
102 = 100
112 = 121
122 = 144
132 = 169
142 = 196
152 = 225
162 = 256
172 = 289
182 = 324
192 = 361
202 = 400
3. 4.16739
16
4.
24
A. Definitions
An expression of the form
radicand and
3
5
a
i
i
r
is a radical expression where
r
is the
is the index. Name the index and the radicand in the following:
( x  y)
abc
7
5x 2
Radical expressions can be simplified using the properties:
n
an  a
Examples:
25 =
196  (___) 2 = _____
(5) 2 = _____
 144 = _______
49 = _________
-
36 =___
64 = _________
Review examples - Simplify the following.
1.
4
2.
4.  81
3. - 225
361
5.
289
B. To simplify radical expressions involving factors, use the property of multiplication with
ab  a  b
radicals:
Examples:
48  16  3  4 3
28  4  7  2 7
Use this property to simplify the following expressions.
1.
20
2.
60
75
4.
90
5.
c.  450
d.
2232
e.  2250
3.
300
You Try:
a.
98
b.
72
C. Adding and Subtracting Radical Expressions.
If the terms have the same radicand just combine their coefficients.
1. 3 5  7 5
2. 2 11  7 11
3.
6 3 6 9 6
4.  3 3  3  8 3
If the terms do NOT have the same radicand, simplify the radicals and then combine those that have the same
radicand.
1. 5 3  12
2. 2 5  3 20
3. 2 7  5 28
4. 13 2  2 18  5 32
You Try:
a. 6 5  2 5
b. 3 98  7 2  18
c.  5 27  4 48
a  b  ab
D. Multiplying Radical expressions.
Multiply the radicands together and then simplify.
1.
3  27
You Try:
a.
3 8
2.
3 7
b.
3.
6  10
5  15
4.
c.
11  33
3 6
E. To simplify radical expressions involving quotients, use the property of division with radicals:
a
a

b
b
(Note: This property works for any index)
Use this property to simplify the following expressions.
1.
49
16
2.
8
2
3.
136
8
4.
You Try:
a.
8
4
b.
125
5
c.
96
2
75
3
F. If a radical remains in the denominator we “agree” to rationalize the denominator as follows:
Multiply the numerator and denominator by the radical in the denominator.
Rationalize the denominator.
1.
5
2
2.
5
12
3.
5
10
4.
3
3
5.
8
8
6.
3
10
b)
7
7
c)
You try:
a)
2
10
25
81