chapter 4 radicals/exponents. §4.2 irrational numbers
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Real Numbers: All Numbers Positive & Negative
Numbers with a decimal representation that neither terminates nor repeats Negative and Positive Numbers Cannot be written as a fraction.
Ex: π, √2 , √-50
Numbers with a decimal representation that terminates or repeats Negative and Positive Numbers Can be written as a fraction.
Ex: 5.3, -16.381, ¾, √ , 0.82
Irrational Numbers:
Rational Numbers:
Negative and Positive whole numbers No DecimalsEx: -4, -3, -2, -1, 0, 1, 2, 3, 4
Integers:
3
9 64
Only positive numbers. No decimals or fractions Includes 0
Ex: 0, 1, 2, 3, 4, ….
Only positive numbers No decimals or fractions Does not include 0.
Ex: 1, 2, 3, 4, 5, 6, ….
Whole Numbers:
Natural Numbers:
Label the following as Rational or Irrational:
1. 2. 3.
4. 5. 6.
Irrational = 4 Rational
Irrational
= 1.2 Rational
= ⅔ Rational
Irrational
44.1
Types of Numbers Real Irrational Rational Integer Whole Natural
-5
0
0.12112111…
-0.75
3
2
132
1
Put a check mark in the box that describes the number given. (Some numbers may require more than one check.)
What type of number is ?36
I Natural
II Integer
III Rational
IV Irrational
= -6Natural #s are Positive
Irrational #s are non-terminating/ non-repeating decimals
D) II & III
Example 2: Write each radical in simplest form, if possible.
a)
b)
c)
2 20
2 10
2 5
3 40
3 52
2 13
26Can not simplify
4 32
2 16
2 8
2 44 22
2 2
Review of Exponent Laws
Law #1 – When multiplying variables we ______the exponents
1)
2)
3)
4)
ADD
)( nmnm xxx
xx4 14 xx 5x
Why do we add the exponents?
(x·x·x·x)(x) = x·x·x·x·x = x5
36 22
65 aa
310 xx
Must have the same
Base
)3(62 32
65a 1a a
)3(10 x 13x
Review of Exponent Laws
Law #2 – When dividing variables we __________ the exponents.
1)
2)
3)
4)
SUBTRACT
)( nmnm xxx 26 xx )26( x 4x
Why do we subtract the exponents?
x·x·x·x·x·x x·x = x·x·x·x = x4
7
10
x
x
3
9
5
5
510 xx
Must have the same
Base
710x 3x
3)9(5 125
)5(10 x 15x
Review of Exponent Laws
Law #3 & #4
1)
2)
3)
4)
5)
nmnm xx )(4)(x 4x Why?
(x3)2 = (x·x·x)(x·x·x) = x6
62 )(b
35 )( a
23)(xy
62b 12b
)3)(5( a 15a
2321 yx 62 yx
nnn yxxy )(
(xy)3 = (xy)(xy)(xy)
= x3y3
= x·y·x·y·x·y= x·x·x·y·y·y
23 )6( ba 21232)6( ba 2636 ba
Review of Exponent LawsLaw #5
1)
2)
3)
4)
n
nn
y
x
y
x
4
y
x4
4
y
x
6
3
2
y
x
3
2
32
ab
ba
23
26
)3(
)3(
63
62
y
x18
12
y
x
3231
3332
ba
ba63
96
ba
ba
23
26
3
3
6936 ba33ba
6
12
3
3 6123 63 729
Why?
xy( )3
= xy( ) x
y( ) xy( )
=x x∙
x∙y y∙
y∙=x3
y3
Review of Exponent Laws
Radical Form
Power Form
Law #6
Example 1: Evaluate each power without using a calculator
a)
b)
c)
d)
mnn mn
m
xorxx
31
27 3 27 2
1
49 49 7
3 x 3
1
x
nn
m
mx
1
3
3
1
64 3 64
2
1
9
4
2
1
2
1
9
4
9
44 3
2
Example 2: Write the following in radical form.
a)
b)
c)
d)
2
1
x x 31
x 3 x
3
2
27 3 227
2
3
16
9
2
3
2
3
16
9 3
3
16
9
23 27or3
3
4
3
64
27
Example 3: Write the following in power form.
a)
b)
c)
d)
x 2
1
x 6 x 6
1
x
n
m
x
6 3x 6
3
x 3 225 3
2
25
2
1
xor
Example 4: Evaluate
a)
b)
c)
d)
2
3
04.0 304.0 3
4
27 43 27
4.0)32( 10
4
)32( 4.1)8.1( 10
14
)8.1(
5
2
)32(
32.0008.0
43 81
25 322)2(
4
5
7
)8.1(
Calculator: 1.8 ^ (7 ÷ 5)
277.2
Biologists use the formula to estimate the brain mass, b kilograms, of a mammal with body mass m kilograms. Estimate the brain mass of each animal
a) a husky with body mass 27 kg
b) a polar bear with a body mass of 200 Kg
3
2
01.0 mb
32
2701.0b 23 2701.0 2)3(01.0 )9(01.0 Kg09.0
32
20001.0b Kg342.0
Calculator: 0.01 x (200 ^ (2 ÷ 3))
a b/c or
Review of Exponent Laws
Law #7 – Anything (number or variable) with a power of zero will
equal 1.
1)
2)
3)
10 a0x 1
06
0)(xy
1
00 yx 11 1
Review of Exponent Laws
Law #8 If there is a negative power
flip it to opposite side of the fraction.
(Reciprocal)
1)
2)
3)
4)
nn
aa
1
2a 2
1
a 2
1
y
3
1
y
x
2y
x
y3 )3)(2()3)(1( yx63 yx
321 )( yx
6
3
y
x
Review of Exponent Laws
Law #8 If there is a negative power
flip it to opposite side of fraction.
(Reciprocal)
5)
6)
nn
aa
1
4
3
2
y
x
3
24
2
yx
yx
)4)(3(
)4)(2(
y
x12
8
y
x8
12
x
y
)3)(2()3)(4(
3)3)(2(
yx
yx612
36
yx
yx 36 yx
Example 1: Evaluate each power
a)
b)
c)
23 23
1 3
4
3
3
3
4
)3(
43.0 46.123
9
1
3
3
)3(
4
27
64
Use a Calculator
0.3 ^ -4 =
Example 2: Evaluate each power without using a calculator
a)
b)
c)
3
2
83
2
8
1 2
3
16
9
2
3
2
3
16
9
2
1
36
25
2
1
2
1
36
25
23 8
1
2
3
2
3
9
16
3
3
9
1622
1
4
1
3
3
3
4
27
64
2
1
2
1
25
36
25
36
5
6
Palaeontologists use measurements from fossilized dinosaur tracks and the formula to estimate the speed at which the dinosaur travelled. In the formula, v is the speed in metres per second, s is the distance between successive footprints of the same foot, and f is the foot length in metres.Use the measurements in the diagram to estimate the speed of the dinosaur.
6
7
3
5
155.0
fsv
6
7
3
5
155.0
fsv
3
5
)00.1(155.0 6
7
)25.0(
sm /781.0Calculator: 0.155 x (1 ^ (5 ÷ 3)) x (0.25 ^ (-7 ÷ 6))
a b/c or
Example 1: Simplify by writing as a single power.
a) b)53 3.03.0
533.0 23.0
3224
2
3
2
3
8
2
3 6
2
3
68
2
3
2
2
3
2
3
2
9
4
Single Power
Example 1: Simplify by writing as a single power.
c) d)2
43
4.1
)4.1)(4.1(
2
7
4.1
4.1
94.1
6
3
5
3
1
3
2
77
7
3
56
3
16
3
26
77
7
102
4
77
7
12
4
7
7 1247
87 87
1
Example 2: Simplify
a) b)))(( 4223 yxyx
4223 yx
22
35
2
10ba
ba
25 yx
2
5
y
x
5 3a 5b
Can’t have negative powers in answer
Example 2: Simplify
c) d)3
163 )8( ba
3
6
3
3
3
1
8 ba
12
122
3
yxyx
213 8 ba22ab
122
1
2
3
yx
12
4
yx
yx2
Simplify each of the following (No Calculators!)
e) f)3
12
3
22
2
4
ba
ba
2
2
1
2
1525
100
ba
a
22a 3
1
3
2
b2
1
2
1
2
15
2
1
2
1
2
1
25
100
ba
a
4
1
2
5
2
1
5
10
ba
a
4
1
2
5
2
1
2
b
a
4
1
2
4
2 ba
2
4
1
2
a
b
3
142 ba
Can’t have negative powers in answer
4
3
1
2
a
b
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