Notes8th Grade

Pre-Algebra

McDowell

Exponents 9/11

Exponents Show repeated multiplication

baseexponent

The number being multiplied

The number of times to multiply the base

Base

Exponent

Example 2³

2 x 2 x 2

4 x 2

8

Example (-2)²

-2²

-2 x –2

4

-1 x 2²-1 x 2 x 2-1 x 4

-4

Examples (12 – 3)² (2² - 1²)

(-a)³ for a = -3

5(2h² – 4)³ for h = 3

Number Sets 9/14Whole

Numbers0, 1, 2, 3, . . .

for short

Also known as the counting numbers

Natural

Numbers

1, 2, 3, 4, . . .

Integers Positive and negative whole numbers

for short

. . . –2, -1, 0, 1, 2, . . .

Rational

NumbersNumbers that can be written as fractions

for short

½, ¾, -¼, 1.6, 8, -5.92

You Try Copy and fill in the Venn Diagram that compares Whole Numbers, Natural Numbers, Integers, and Rational Numbers

Whole #s

Prime

Numbers

Integers greater than one with two positive factors

1 and the original number

Integers greater than one with more than two positive factors

Composite

Numbers

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, . . .

4, 6, 8, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, . . .

Factor Trees

A way to factor a number into its prime factors

Steps

Is the number even or odd?If even: divide by 2If odd: divide by 3, 5, 7, 11,

13 or another prime numberWrite down the prime factor and

the new numberIs the new number prime or composite?

Is the number prime or composite?

If Composite:If prime: you’re done

Example Find the prime factors of

99even or odd

divide by 3

3 33even or odd

divide by 3

prime or composite

prime or composite

3 11 prime or composite

The prime factors of 99: 3, 3, 11

Example Find the prime factors of

12even or odd

divide by 2

2 6even or odd

divide by 2

prime or composite

prime or composite

2 3 prime or composite

The prime factors of 12: 2, 2, 3

You Try Find the prime factors of

1. 8

2. 15

3. 82

4. 124

5. 26

GCF 9/15

GCF Greatest Common Factor

the largest factor two or more numbers have in common.

Steps toFindingGCF

1. Find the prime factors of each number or expression

3. Pick out the prime factors that match

2. Compare the factors

4. Multiply them together

Example Find the GCF of 126 and 150

150126

2 63

3

2

5

75

21 15

5 3

The common factors are 2, 3

3 7

2 x 3The GCF of 126 and 150 is 6

Example

Find the GCF of 24x4 and 16x3

16xxx24xxxx

2 12

2

2

2

8

6 4

2 2

The common factors are 2, 2, 2, x, x, x

2 3

2(2)(2)xxxThe GCF is 8x3

You Try Work Book

P 62

# 2 - 24 even

Simplifying Fractions 9/16

Simplest form

When the numerator and denominator have no common factors

Simplifying fractions

1. Find the GCF between the numerator and denominator

2. Divide both the numerator and denominator of the fraction by that GCF

Example Simplify 2852

28s Prime factors: 2, 2, 752s Prime factors: 2, 2, 13

Use a factor tree to find the prime factors of both numbers and then the GCF

GCF: 2 x 24

2852

4

4

= 7 13

Example Simplify 12a5b6

18a2b8

12s Prime factors: 2, 2, 318s Prime factors: 2, 3, 3

Use a factor tree to find the prime factors of both numbers and then the GCF

GCF: 2 x 36

1218

6

6

= 2aaaaabbbbbb 3aabbbbbbbb

2aaa 3bb

2a3

3b2

You Try Write each fraction in simplest form1. 27 30

2. 15x2y 45xy3

½ and 2/4 are equivalent fractions

Fractions that represent the same amount

Equivalent fractions

Making

Equivalent

Fractions

1. Pick a number

2. Multiply the numerator and denominator by that same number

58

x 3x 3

= 15 24

You Try Find 3 equivalent fractions to

611

Are the

Fractions

equivalent?

1. Simplify each fraction

2. Compare the simplified fraction

3. If they are the same then they are equivalent

You try Work Book

p 49 #1-17 odd

Least common Denominator 9/17

Common

DenominatorWhen fractions have the same denominator

Steps to

Making

Common

Denominators

1. Find the LCM of all the denominators

2. Turn the denominator of each fraction into that LCM using multiplication

Remember: what ever you multiply by on the bottom, you have to multiply by on the top!

Example Make each fraction have a common denominator

5/6, 4/9 Find the LCM of 6 and 9

6 12 18 24 30 36 42 489 18 27 36 45 64 73 82

Multiply to change each denominator to 18

5 x 36 x 3

= 15 18

= 8 18

4 x 29 x 2

You try What are the least common denominators?

1. ¼ and 1/3

2. 5/7 and 13/12

Comparing

And

Ordering

fractions

Manipulate the fractions so each has the same denominator

Compare/order the fractions using the numerators (the denominators are the same)

You try Order the rational numbers from least to greatest

1. 8/15, 6/13, 5/9, 4/7

2. -2/3, ½, 4/7, -4/5

Graph each group of rational numbers on a number line

-1 0 1

Evaluating fractions

Plug and chug

Substitute in the values for the variables then chug chug chug out the answer in simplest form

ExampleEvaluate x(xy – 8) for x = 3 and y = 9

60

Plug3(3•9 – 8) 60

Chug Remember Sally

3(27 – 8) 60

3(19) 60

1920

3 3

You try Workbook

p 68

# 1-17 odd, 18

Exponents and Multiplication9/18

The long way

25 • 23

(2 • 2 • 2 • 2 • 2) • (2 • 2 • 2)

28

expand

Convert back to exponential form

The short way

Same bases so we can add the exponents

25 • 23

25+3

28

Simplify

Multiplying

Powers

With the

Same base

Works for numbers and variables

When same base powers are multiplied, just add the exponents

Rememberbaseexponent

Examples x2x2x2

x2+2+2

x6

32y5 • 34y10

32 • 34y5y10 Associative Property

32+4y5+10 Add exponents

36y15

You Try 1. x5x7

2. 74a8 • 7a11

A Parisian mathematician, Nicolas Chuquet, who is credited with the first use exponents and with naming large numbers (billion, trillion, etc.)

Raising a power to a power 9/18

The long way

(x2)3

(x • x) • (x • x) • (x • x)

x6

expand

Convert back to exponential form

x2 • x2 • x2

The short way

Multiply the exponents

x6

(x2)3

You try 1. (x6)7

2. (x8)5

Exponent means “out of place” in Latin

Micheal Stifel named exponents—he was German, a monk, a mathematics professor. He was once arrested for predicting the end of the world once it was proven he was wrong.

You try Workbook

p 68

# 1-17 odd, 18

Exponent Rules 9/21

Exponents

Rules

x0 = 1for x 0

10980 = 1

(-23)0 = 1

Everything raised to the zero power is 1(except zero)

Exponent

Rules

Negative exponents mean the exponential is on the wrong side of the fraction bar

x-2 = 1 x2

Make that power happy by moving it to the other side of the fraction bar

Examples

Simplify

a-3 = 1a3

1y-5 = y5

b-10 =2-2

22

b10

You Try Simplify

1. a-12

2. 1 x-7

3. c-10

c2d-3

Division and Exponents 9/21

The long way

x6

x9

1x3

expand

Cross out pairs x x x x x x x x x x x x x x x

The short way

Subtract the exponents

Top minus bottom

x6-9

x6

x9

Simplify

1x3

x-3 Make all exponents positive

9 is bigger than 6 so it makes sense that the x is in the denominator

Examples Simplify

45x4y7

9x6y3

You try 1. x5

x4

2. a10

a12

3. 16a2b4

8a5b2

Scientific Notation 9/22

Powers

Of

Ten

Factors 10 10x10 10x10x10 10x10x10x10

Product 10 100 1,000 10,000

Power 101 102 103 104

# of 0s 1 2 3 4

Factors 1

10

1

10x10

1

10x10x10

1

10x10x10x10

Product 0.1 0.01 0.001 0.0001

Power 10-1 10-2 10-3 10-4

# of 0s

After the decimal

0 1 2 3

Scientific

Notation

Looks like:

2.4 x 104

A short way to write really big or really small numbers using factors

The other factor will be less than 10 but greater than one

1 < factor < 10

And will usually have a decimal

One factor will always be a power of ten: 10n

The first factor tells us what the number looks like

The exponent on the ten tells us how many places to move the decimal point

A positive exponent moves the decimal to the right

A negative exponent moves the decimal to the left

Makes the number bigger

Makes the number smaller

4.6 x 106

4600000

Example

Move the decimal 6 hops to the right

4.600000 Rewrite

Convert between scientific notation and expanded notation

Write in expanded notation

1. 2.3 x 10-3

2. 5.76 x 107

Answers

1. 0.0023

2. 57,600,000

You Try

13,700,000

1.3 x 107

Example

Figure out how many hops it takes to get a factor between 1 and 10

1.3,700,000 Rewrite: the number of hops is your exponent

Convert between expanded notation and scientific notation

If you hop left the exponent will be positive---the number is bigger than 0

If you hop right the exponent will be negative---the number is less than zero

Write in scientific notation

1. 340,000,000

2. 0.000982

Answers

1. 3.4 x 108

2. 9.82 x 10-4

You Try