5.4 the Irrational Numbers and the Real Number System______

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. 16 3. 4.16739 4. 24

MEMORIZE A. Definitions Perfect “Squares” An expression of the form i r is a radical expression where r is the 12 = 1 22 = 4 radicand and i is the index. Name the index and the radicand in the following: 32 = 9 2 = 4 16 3 a 5 abc (x  y) 7 5x 2 52 = 25 2 6 = 36 Radical expressions can be simplified using the properties: 72 = 49 2 8 = 64 n n 92 = 81 a  a 102 = 100 112 = 121 Examples: 122 = 144 2 13 = 169 25 = (5) 2 = _____ 196  (___) 2 = _____ 36 =___ 142 = 196

152 = 225 162 = 256  144 = ______49 = ______- 64 = ______172 = 289 182 = 324 Review examples - Simplify the following. 192 = 361 202 = 400 1. 4 2. 361 3. - 225 4.  81 5. 289

B. To simplify radical expressions involving factors, use the property of multiplication with radicals: ab  a  b

Examples: 28  4  7  2 7 48  16  3  4 3

Use this property to simplify the following expressions.

1. 20 2. 60 3. 75 4. 90 5. 300

You Try: a. 98 b. 72 c.  450 d. 2232 e.  2250 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

D. Multiplying Radical expressions. a  b  ab

Multiply the radicands together and then simplify.

1. 3  27 2. 3  7 3. 6  10 4. 11  33

You Try: a. 3  8 b. 5  15 c. 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.

49 8 136 75 1. 2. 3. 4. 16 2 8 3

You Try:

8 125 96 a. b. c. 4 5 2

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.

5 5 5 1. 2. 3. 2 12 10

3 8 3 4. 5. 6. 3 8 10

You try:

2 7 25 a) b) c) 10 7 81