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Notes 8th Grade Pre-Algebra McDowell Exponents Exponents 9/11 Show repeated multiplication baseexponent Base The number being multiplied Exponent The number of times to multiply the base Example 2³ 2x2x2 4x2 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 Whole Numbers Natural Numbers 9/14 0, 1, 2, 3, . . . for short Also known as the counting numbers 1, 2, 3, 4, . . . Integers Positive and negative whole numbers for short . . . –2, -1, 0, 1, 2, . . . Rational Numbers that can be written as Numbers 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 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? 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 1. 8 2. 15 3. 82 4. 124 5. 26 GCF GCF 9/15 Greatest Common Factor the largest factor two or more numbers have in common. Steps to Finding GCF 1. Find the prime factors of each number or expression 2. Compare the factors 3. Pick out the prime factors that match 4. Multiply them together Example Find the GCF of 126 and 150 126 2 150 63 3 75 2 21 15 5 5 3 7 The common factors are 2, 3 2x3 The GCF of 126 and 150 is 6 3 Exampl e Find the GCF of 24x4 and 16x3 24xxxx 2 16xxx 12 6 2 2 8 2 4 2 3 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 Simplest form When the numerator and denominator have no common factors 9/16 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 52s Prime factors: 2, 2, 13 28 4 = 7 52 4 13 Use a factor tree to find the prime factors of both numbers and then the GCF GCF: 2 x 2 4 Example Simplify 12a5b6 18a2b8 Use a factor tree 12s Prime factors: 2, 2, 3 18s Prime factors: 2, 3, 3 12 6 = 2aaaaabbbbbb 18 6 3aabbbbbbbb 2aaa 3bb 2a3 3b2 to find the prime factors of both numbers and then the GCF GCF: 2 x 3 6 You Try Write each fraction in simplest form 1. 27 30 2. 15x2y 45xy3 Equivalent fractions Fractions that represent the same amount ½ and 2/4 are equivalent fractions Making Equivalen t Fractions 1. Pick a number 2. Multiply the numerator and denominator by that same number 5 x 3 = 15 8 x 3 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 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! 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 5 x 3 = 15 each denominator 6 x 3 18 to 18 4x2=8 9 x 2 18 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 Example Evaluate x(xy – 8) for x = 3 and y = 9 60 3(3•9 – 8) 60 Plug 3(27 – 8) 60 Chug Remember Sally 3(19) 3 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 28 Convert back to exponential form The short way 25 • 23 Same bases so we can add the exponents 25+3 Simplify 28 Multiplying Works for numbers and variables Powers When same base powers are With the multiplied, just add the Same base exponents Remember baseexponent Example s x2x2x2 x2+2+2 x6 32y5 • 34y10 32 • 34y5y10 Associative Property 32+4y5+10 36y15 Add exponents 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 The long way (x2)3 x2 • x2 • x2 9/18 expand (x • x) • (x • x) • (x • x) x6 Convert back to exponential form The short way (x2)3 x6 Multiply the exponents 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 Exponents Rules 9/21 Everything raised to the zero power is 1(except zero) 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 Example s Simplify a-3 1 = 3 a 1 5 = y y-5 b-10 = 22 2-2 b10 You Try Simplify 1. a-12 2. 1 x-7 3. c-10 c2d-3 Division and Exponents The long way 9/21 x6 x9 expand xxxxxx xxxxxxxxx Cross out pairs 1 x3 The short way x6 x9 Subtract the exponents x6-9 Simplify x-3 Make all exponents positive 1 x3 Top minus bottom 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 Powers Of Ten 9/22 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 1 1 1 10 10x10 10x10x10 10x10x10x10 Product 0.1 0.01 0.001 0.0001 Power 10- 10-2 10-3 10-4 0 2 3 1 # of 0s After the decimal 1 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 Example Convert between scientific notation and expanded notation 4.6 x 106 Move the decimal 6 hops to the right 4.600000 Rewrite 4600000 You Try Write in expanded notation 1. 2.3 x 10-3 2. 5.76 x 107 Answers 1. 0.0023 2. 57,600,000 Example Convert between expanded notation and scientific notation 13,700,000 Figure out how many hops it takes to get a factor between 1 and 10 1.3,700,000 1.3 x 107 Rewrite: the number of hops is your exponent 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 1. 340,000,000 2. 0.000982 Answers 1. 3.4 x 108 2. 9.82 x 10-4