chapter 4 number theory and fractions. 4-1: exponents iwbat write and evaluate exponential...

CHAPTER 4

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4-1: E

XPONENTS

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VOCABULARY

• Exponent – Tells how many times a repeated factor is to be multiplied.

33Power – How many times

to multiply

Base – The factor to be multiplied

3 ∙3 ∙327

WRITING EXPONENTS

2×2×2

The factor is the base. The number of times the factor appears is the exponent.

23

0.8 ∙0.8 ∙0.8 ∙0.8

0.84

𝑏 ∙𝑏 ∙𝑏 ∙𝑏 ∙𝑏∙𝑏

𝑏6

EVALUATING EXPONENTS

43

1. Write out what the exponent represents

4 ∙4 ∙ 4

2. Multiply. Write out each multiplication step

16 ∙4

64

EXAMPLES

1)

7 ∙749

2)

6 ∙6 ∙636 ∙6

216

3)

0.5 ∙0.5 ∙0.5 ∙0.50.25 ∙0.250.0625

4)

4

EVALUATING WITH SUBSTITUTING

Evaluate when n = 5

𝑛2−21) Rewrite the problem with the substitution.

2) Substitute for the variable. 52−2

3) Evaluate using order of operations.

25−2

23

EXAMPLES

Evaluate each expression when n = 5.

1)

(5−3)4

24

2 ∙2∙2 ∙24 ∙416

2)

2 ∙52

2 ∙25

50

WORD PROBLEM

A certain cell doubles every hour. If you begin with one cell, at the end of 1 hour there are 2 cells, at the end of 2 hours, there are or 4 cells, and so on. After 6 hours, how many cells will there be?

Hour 1 = 2

Hour 2 = = 4

Hour 3 = = 8

Hour 6 =

64 cells in 6 hours.

4-2: P

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DIVISIBILITY RULES

A number is divisible by . . . .

• 2 if the last digit is even (ends in 0, 2, 4, 6, 8)

• 3 if the sum of the digits is divisible by 3

• 4 if the number formed by the last two digits are divisible by 4

• 5 if the last digit is a 0 or 5

• 6 if the number is divisible by both 2 and 3

• 9 if the sum of the digits is divisible by 9

• 10 if the last digit is a 0

EXAMPLES

Tell if 3,742 is divisible by 2, 3, 4, 5, 6, 9, or 10.

1) Test the 2s. Is the number even?

yes

2) Test the 3s. Find the digital root.

3+7+4+2 = 16

Is 16 divisible by 3? No

3) Test the 4s. Look at the last two numbers.

Is 42 divisible by 4?

No

4) Test the 5s. Does it end in 0 or 5?

No.

5) Test the 6s. Can it be divisible by 2 and 3?

Just 2 and not 3. No

6) Test the 9s. Find the digital root.

3+7+4+2 = 16 = 1+6 = 7

No

7) Test the 10s. Does it end in 0?

No

3,742 is only divisible by 2

EXAMPLES

Tell if each number is divisible by 2, 3, 4, 5, 6, 9, 10 or none of these

1) 3,742 2 3 6

2) 5,310 2 5 9 10

3) 47,388 2 3 6

4) 9,999 3 9

VOCABULARY

• Prime Number – A whole number with only two factors, 1 and itself

• Composite Number – A whole number with more than two factors.

PRIME FACTORIZATION

Breaking down a number so that it is represented as a product of prime numbers

Use the factor tree method to find the prime factorization.

4-3: R

EASONABLE

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4-4: G

REATEST

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GREATEST COMMON FACTOR (GCF)

Two methods to finding the GCF

1. Factor Rainbow

2. Using Prime Factorizing

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4-6: L

EAST COMMON

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LEAST COMMON MULTIPLE (LCM)

Use the same methods of finding the GCF to find the LCM, with one major difference.

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FINDING EQUIVALENT FRACTIONS

M U LT I P LY I N G

Multiply both the numerator and the denominator

D I V I D I N G

Divide both the numerator and the denominator by the GCF

SIMPLEST FORM IS WHEN THE GCF IS 1 (YOU CAN’T DIVIDE ANYMORE)

Write in simplest form

1) Find the GCF of 12 and 42 12 – 1 2 3 4 6 1242 – 1 2 3 6 7 14 21 42GCF = 6

2) Divide the numerator and denominator 12

42÷6÷6

¿27

EXAMPLES

1)

Write each fraction in simplest form

12 – 1 2 3 4 6 1222 – 1 2 11 22

GCF = 2

1222÷2÷2¿

611

2)

14 – 1 2 7 14

28 – 1 2 4 7 14 28GCF = 14

1428÷14÷14

¿12

FINDING THE LEAST COMMON DENOMINATORLeast Common Denominator (LCD) – The common denominator

(or multiple) of two or more fractions.

The LCD is the same thing as the LCM but now used with fractions.

REWRITE AND AS FRACTIONS WITH THE SAME DENOMINATOR

1) Find the LCM of the denominator 4 – 4 8 12 16 20

6 – 6 12

LCM = 12

2) Use the LCM as the new denominator for each fraction

34=❑12

56= ❑12

3) “What you do to the bottom, you do to the top”

34×3×3

¿912

56×2×2

¿1012

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EXAMPLES

Use the LCD to write each set of fractions with the same denominator.

1)

20 – 20 40 60

LCM: 60

2)

30 - 30

LCM: 30

3)

9 – 9 18 27 36 45 54 63 72LCM: 72

9672,4572,5672

4-8: C

OMPARIN

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ORDERING F

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VOCABULARY

Improper Fraction – A fraction where the numerator is larger than the denominator. They can be rewritten as mixed numbers.

Ex:

Mixed Number – A number that is made of a whole number and a fraction.

Ex: 1

IMPROPER FRACTIONS AND MIXED NUMBERSI M P R O P E R F R A C T I O N T O M I X E D N U M B E R

Divide the denominator into the numerator.

The quotient is the whole number.

The remainder is the new numerator.

The divisor is the new denominator

M I X E D N U M B E R S T O I M P R O P E R F R A C T I O N S

Multiply the whole number and the denominator.

Add the numerator. The sum is the new numerator.

Use the same denominator.

IMPROPER FRACTION TO MIXED NUMBER

1)

7÷3=2𝑟 1

213

2)

22÷3=7𝑟 1

713

3)

17÷8=2𝑟 1

218

MIXED NUMER TO IMPROPER FRACTION

1)

7×9=6363+4=67677

2)

5×10=5050+3=53535

3)

1×4=44+3=774

COMPARING FRACTIONS

1. Compare whole numbers first.

2. Find the LCD for all fractions.

3. Compare or order from least to greatest.

Compare. Write <, >, or =.

1)

LCD: 8

58

34=68

58<68

2)

LCD: 9

7923=69

79>69

3) 3

LCD: 20

45=1620

34=1520

31620

>31520

ORDERING FRACTIONS

1. Rewrite all mixed numbers as improper fractions

2. Find the LCD for all fractions

3. Compare

4. Order from least to greatest

5. Rewrite using original fractions and mixed numbers

6. Place the fractions on a number line

Arrange in order from least to greatest and place them on a number line.

1)

134=74

LCD: 8

78,128,148,178

78,32,134,178

EXAMPLES

Arrange in order from least to greatest.

1)

LCD: 56

2156,2856,4056

38,12,57

2)

LCD: 18

618,818,1518

13,49,56

3)

LCD: 16

14=416

5416,51116,6

514,51116,6

4-9: R

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CONVERTING FRACTIONS AND DECIMALS

F R A C T I O N T O D E C I M A L

• Divide the numerator by the denominator

• If there is a whole number, place the whole number in front of the decimal

• If the decimal repeats, round to the nearest hundredths

D E C I M A L T O F R A C T I O N

• Place the decimal digits over the place value

• Rewrite in simplest form

• If there is a whole number, place the whole number in front of the fraction

EXAMPLES

Write each fraction as a decimal.

1)

3÷ 4=0.750.75

2)

2÷3=0.66660.67

3)

3÷20=0.154.15

Write each decimal as a fraction in simplest form.

1) 0.6

Tenths place = 10

610

¿35

35

2) 0.08

Hundredths place = 1008100

¿225

225

3) 5.11

Hundredths place = 100

11100

511100

4-10: U

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