8th Grade Pre-Algebra McDowell Notes 8th Grade Pre-Algebra McDowell Chapter 4

Exponents 9/11 Base Exponent Exponents Show repeated multiplication The number being multiplied Exponent The number of times to multiply the base

Example 2³ 2 x 2 x 2 4 x 2 8

Example (-2)² -2 x –2 4 -2² -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 9/14 Whole Numbers 0, 1, 2, 3, . . . Natural for short 0, 1, 2, 3, . . . Natural Numbers for short Also known as the counting numbers 1, 2, 3, 4, . . .

Numbers that can be written as fractions Integers Positive and negative whole numbers for short . . . –2, -1, 0, 1, 2, . . . Rational Numbers Numbers 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

Integers greater than one with more than two positive factors Integers greater than one with two positive factors 1 and the original number Prime Numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, . . . Composite Numbers Integers greater than one with more than two positive factors 4, 6, 8, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, . . .

Factor Trees Steps A way to factor a number into its prime factors Is the number prime or composite? Steps If prime: you’re done If Composite: Is the number even or odd? If even: divide by 2 If odd: divide by 3, 5, 7, 11, 13 or another prime number Write down the prime factor and the new number Is the new number prime or composite?

Example Find the prime factors of 99 prime or composite even or odd divide by 3 3 33 prime or composite even or odd divide by 3 3 11 prime or composite The prime factors of 99: 3, 3, 11

Example Find the prime factors of 12 prime or composite even or odd divide by 2 2 6 prime or composite even or odd divide by 2 2 3 prime or composite The prime factors of 12: 2, 2, 3

You Try Find the prime factors of 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.

1. Find the prime factors of each number or expression Steps to Finding GCF 2. Compare the factors 3. Pick out the prime factors that match 4. Multiply them together

126 150 2 63 75 2 15 21 5 3 5 3 3 7 The common factors are 2, 3 2 x 3 Find the GCF of 126 and 150 Example 126 150 2 63 75 2 15 21 5 3 5 3 3 7 The common factors are 2, 3 2 x 3 The GCF of 126 and 150 is 6

Example Find the GCF of 24x4 and 16x3 24xxxx 16xxx 2 12 8 2 4 6 2 2 2 The common factors are 2, 2, 2, x, x, x 2(2)(2)xxx The 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 28 52 28s Prime factors: 2, 2, 7 Use a factor tree to find the prime factors of both numbers and then the GCF 28s Prime factors: 2, 2, 7 52s Prime factors: 2, 2, 13 GCF: 2 x 2 4 28 52  4 = 7 13

Example Simplify 12a5b6 18a2b8 12s Prime factors: 2, 2, 3 Use a factor tree to find the prime factors of both numbers and then the GCF 12s Prime factors: 2, 2, 3 18s Prime factors: 2, 3, 3 12 18  6 = 2aaaaabbbbbb 3aabbbbbbbb GCF: 2 x 3 6 2aaa 3bb 2a3 3b2

You Try Write each fraction in simplest form 27 30 15x2y 45xy3

Fractions that represent the same amount Equivalent fractions Fractions that represent the same amount ½ and 2/4 are equivalent fractions

Making Equivalent Fractions 1. Pick a number 2. Multiply the numerator and denominator by that same number 5 8 x 3 = 15 24

You Try Find 3 equivalent fractions to 6 11

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 When 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!

Make each fraction have a common denominator 5/6, 4/9 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 48 9 18 27 36 45 64 73 82 Multiply to change each denominator to 18 5 x 3 6 x 3 = 15 18 4 x 2 9 x 2 = 8 18

What are the least common denominators? ¼ and 1/3 5/7 and 13/12 You try What are the least common denominators? ¼ and 1/3 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 8/15, 6/13, 5/9, 4/7 -2/3, ½, 4/7, -4/5 Graph each group of rational numbers on a number line -1 1

Evaluating fractions Plug and chug Substitute in the values for the variables then chug chug chug out the answer in simplest form

Example Plug Chug Remember Sally Evaluate x(xy – 8) for x = 3 and y = 9 60 Example Plug 3(3•9 – 8) 60 Chug Remember Sally 3(27 – 8) 60 3(19) 60 3 19 20

You try Workbook p 68 # 1-17 odd, 18

Exponents and Multiplication 9/18 The long way 25 • 23 (2 • 2 • 2 • 2 • 2) • (2 • 2 • 2) expand Convert back to exponential form 28

The short way 25 • 23 25+3 28 Same bases so we can add the exponents Simplify 28

Multiplying Powers With the Same base Works for numbers and variables When same base powers are multiplied, just add the exponents Remember baseexponent

Examples x2x2x2 x2+2+2 x6 32y5 • 34y10 32 • 34y5y10 Associative Property 32+4y5+10 Add exponents 36y15

You Try x5x7 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 x2 • x2 • x2 expand (x • x) • (x • x) • (x • x) Convert back to exponential form x6

The short way Multiply the exponents (x2)3 x6

Exponent means “out of place” in Latin You try (x6)7 (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 Everything raised to the zero power is 1(except zero) Exponents Rules x0 = 1for x  0 10980 = 1 (-23)0 = 1

Exponent Rules Negative exponents mean the exponential is on the wrong side of the fraction bar Make that power happy by moving it to the other side of the fraction bar x-2 = 1 x2

Examples Simplify 1 a3 a-3 = 1 y-5 = y5 b-10 = 2-2 22 b10

You Try Simplify a-12 1 x-7 3. c-10 c2d-3

Division and Exponents 9/21 The long way expand x x x x x x x x x x x x x x x Cross out pairs 1 x3

The short way x6 x9 x6-9 x-3 1 x3 Subtract the exponents Top minus bottom x6-9 Simplify x-3 Make all exponents positive 1 x3 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 Product 100 1,000 10,000 Power 101 102 103 104 # of 0s 1 2 3 4

Factors 1 10 10x10 10x10x10 10x10x10x10 Product 0.1 0.01 0.001 0.0001 Power 10-1 10-2 10-3 10-4 # of 0s After the decimal 2 3

A short way to write really big or really small numbers using factors Scientific Notation A short way to write really big or really small numbers using factors Looks like: 2.4 x 104

One factor will always be a power of ten: 10n The other factor will be less than 10 but greater than one 1 < factor < 10 And will usually have a decimal

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 Makes the number bigger A negative exponent moves the decimal to the left Makes the number smaller

Move the decimal 6 hops to the right 4.6 x 106 Convert between scientific notation and expanded notation Example Move the decimal 6 hops to the right 4.6 x 106 4.600000 Rewrite 4600000

You Try Write in expanded notation 2.3 x 10-3 5.76 x 107 Answers 0.0023 57,600,000

Convert between expanded notation and scientific notation Example 13,700,000 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 1.3 x 107

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

You Try Write in scientific notation 340,000,000 0.000982 Answers 3.4 x 108 9.82 x 10-4