Chapter 4 Radicals/Exponents. §4.2 Irrational Numbers.

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Chapter 4 Radicals/Exponents

Transcript of Chapter 4 Radicals/Exponents. §4.2 Irrational Numbers.

Chapter 4Radicals/Exponents

§4.2Irrational Numbers

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

§4.3Mixed and Entire Radicals

Simplify each radical

a) 80

2 40

2 20

2 10

2 5

80

22x 54

2

5

Simplify each radical

b)

c) 2 72

2 36

2 18

2 9

3 144

3 3322

2 81

3 27

3 9

4 162

4 233 3

3 182

3 3

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

§4.4Fractional Exponents and Radicals

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

§4.5Negative Exponents and Reciprocals

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

§4.6Applying the Exponent Laws

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

A sphere has a volume of 425 cubic metres. What is the radius of the sphere to the nearest tenth of a metre?

3

3

4rV

3

3

4425 r3· ·3

1275 = 4πr3

4π 4π

1275 = r3

4π √3

√ 3

Calculator: 3√((1275 ÷ (4xπ))=

4.7m = r