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Notes

Notes. Chapter 4. 8 th Grade Pre-Algebra McDowell. Exponents. Exponents 9/11. Show repeated multiplication. base exponent. Base. The number being multiplied. Exponent. The number of times to multiply the base. Example. 2 ³. 2 x 2 x 2. 4 x 2. 8. Example. (-2) ². -2 x –2. 4.

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

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

  3. Example 2³ 2 x 2 x 2 • 4 x 2 8

  4. Example (-2)² • -2 x –2 4 • -2² • -1 x 2² • -1 x 2 x 2 • -1 x 4 • -4

  5. Examples (12 – 3)²  (2² - 1²) • (-a)³ for a = -3 • 5(2h² – 4)³ for h = 3

  6. Whole Numbers Number Sets 9/14 0, 1, 2, 3, . . . Natural Numbers for short Also known as the counting numbers 1, 2, 3, 4, . . .

  7. 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

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

  9. 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, . . .

  10. Factor Trees 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?

  11. 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

  12. 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

  13. You Try Find the prime factors of 8 2. 15 3. 82 4. 124 5. 26

  14. GCF GCF 9/15 Greatest Common Factor the largest factor two or more numbers have in common.

  15. 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

  16. Example Find the GCF of 126 and 150 • 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

  17. Example Find the GCF of 24x4 and 16x3 • 24xxxx • 16xxx • 2 • 12 • 8 • 2 • 4 • 6 • 2 • 2 • 2 • 2 • 2 • 3 • The common factors are 2, 2, 2, x, x, x • 2(2)(2)xxx • The GCF is 8x3

  18. You Try Work Book P 62 # 2 - 24 even

  19. Simplest form Simplifying Fractions 9/16 When the numerator and denominator have no common factors

  20. Simplifying fractions 1. Find the GCF between the numerator and denominator • 2. Divide both the numerator and denominator of the fraction by that GCF

  21. Example Simplify 28 52 • 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 •  4 • = 7 • 13

  22. Example Simplify 12a5b6 18a2b8 • 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 •  6 • = 2aaaaabbbbbb • 3aabbbbbbbb • GCF: 2 x 3 • 6 • 2aaa • 3bb • 2a3 • 3b2

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

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

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

  26. You Try Find 3 equivalent fractions to 6 11

  27. Are the Fractions equivalent? 1. Simplify each fraction • 2. Compare the simplified fraction • 3. If they are the same then they are equivalent

  28. You try Work Book p 49 #1-17 odd

  29. Common Denominator Least common Denominator 9/17 When fractions have the same denominator

  30. 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!

  31. 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

  32. You try What are the least common denominators? ¼ and 1/3 5/7 and 13/12

  33. 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)

  34. 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 0 1

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

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

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

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

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

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

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

  42. 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.)

  43. The long way Raising a power to a power 9/18 (x2)3 x2 • x2 • x2 expand (x • x) • (x • x) • (x • x) Convert back to exponential form x6

  44. The short way Multiply the exponents (x2)3 x6

  45. 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.

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

  47. Exponents Rules Everything raised to the zero power is 1(except zero) Exponent Rules 9/21 • x0 = 1for x  0 • 10980 = 1 • (-23)0 = 1

  48. 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

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

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