1. Chapter 4 Notes

2. 4-1 Divisibility and Factors Divisibility Rules for 2, 5, and 10 An integer is divisible by –2 if it ends in 0, 2, 4, 6, or 8 –5 if it ends in 0 or 5 –10 if it ends in 0

3. Examples Is the first number divisible by the second? 567 by 2 1015 by 5 111,120 by 10

4. Examples - Answers Is the first number divisible by the second? 567 by 2no; it ends in an odd number 1015 by 5yes; it ends in a 5 111,120 by 10yes; it ends in a 0

5. Divisibility rules for 3 and 9 An integer is divisible by –3 if the sum of its digits is divisible by 3 –9 if the sum of its digits is divisible by 9

6. Examples - Answers Is the first number divisible by the second? 567 by 3 yes; 5+6+7=18 and 18 is divisible by 3 1015 by 9no; 1+0+1+5=7 and 7 is NOT divisible by 9

7. Divisibility rules for 4, 6, 8 An integer is divisible by –4 if the number formed by the last 2 digits is divisible by 4 –6 if the number is divisible by BOTH 2 and 3 –8 if the number formed by the last 3 digits is divisible by 8

8. Examples If the first number divisible by the second? 532 by 4yes; 32 is divisible by 4 342 by 6yes; 3+4+2= 9 and is divisible by 3 also 2 is even which is divisible by 2 5,832 by 8 yes; 832 is divisible by 8

9. Divisibility rules for 7 There are no rules for 7. You just need to work it out!!!

10. Finding Factors Factor: a number is a factor of another number if it divides into that number with a remainder of 0 Examples: 21--> 1, 3, 7, 21 31 --> 1, 31 24 --> 1, 2, 3, 4, 6, 8, 12, 24

11. 4-2 Exponents Exponents - show repeated multiplication 2 6 = 2 x 2 x 2 x 2 x 2 x 2 = 64 Base -->2 Exponent --> 6 Value of expression --> 64 a 2 = a x a b 4= b x b x b x b

12. Write the expression using an exponent. (-5)(-5)(-5) -2 x a x b x a 6 x 6 x 6 4 x d x d x c x c

13. Write the expression using an exponent. Answers (-5)(-5)(-5) = (-5) 3 -2 x a x b x a = -2 x a x a x b = -2a 2 b 6 x 6 x 6 = 6 3 4 x d x d x c x c = 4c 2 d 2

14. Examples (-3)(-3)(-3)(-3) -7 x a x a x b Simplify: 10 4 (-5) 4 -5 4

15. Examples - Answers (-3)(-3)(-3)(-3) = (-3) 4 -7 x a x a x b = (-7)a 2 b Simplify: 10 4 = 10 x 10 x 10 x 10 = 10,000 (-5) 4 = (-5) x (-5) x (-5) x (-5) = 625 -5 4 = - (5 x 5 x 5 x 5) = -625

16. Using the Order of Operations Simplify 4(3 + 2) 2 =4(5) 2 =4 x 5 x 5 or 4 x 25 = 100

17. Examples 49 - (4 x 2) 2 2(9 - 4) 2 (-4)(-6) 2 (2) (12 - 3) 2 - (2 2 - 1 2 )

18. Examples - Answers 49 - (4 x 2) 2 = -15 2(9 - 4) 2 = 50 (-4)(-6) 2 (2) = 288 (12 - 3) 2 - (2 2 - 1 2 ) = 78

19. Evaluate each expression c 3 + 4, for c = -6 3(2m + 5) 2, for m = 2

20. Evaluate each expression - Answers c 3 + 4, for c = -6 =(-6) 3 + 4 = -216 + 4 = -212 3(2m + 5) 2, for m = 2 =3(2x2 + 5) 2 =3(9) 2 = 3(81) = 243

21. 4-3 GCF and LCM GCF = Greatest Common Factor LCM = Least Common Multiple Prime Factorization = factor tree Prime number = has exactly 2 factors - 1 and itself Composite number = has more than 2 factors

22. Write the prime factorization 825 34 360 186

23. Write the prime factorization (PF) - Answers 825PF = 3 * 5 2 * 11 34PF = 2 * 17 360PF = 2 3 * 3 2 * 5 186PF = 2 * 3 * 31

24. Relatively prime Two numbers are relatively prime if their GCF is 1 Examples: 8, 17: Yes, because their GCF is 1 7, 35: No, because their GCF is 7

25. Find each GCF and LCM 42, 60 8, 16, 20 180a 2, 210a a 3 b, a 2 b 2

26. Find each GCF and LCM - Answers 42, 60GCF:6, LCM: 420 8, 16, 20GCF: 4, LCM: 160 180a 2, 210aGCF: 30a, LCM:1260a 2 a 3 b, a 2 b 2 GCF: a 2 b, LCM:a 3 b 2

27. 4-4 Simplifying Fractions Finding Equivalent Fractions = multiply or divide the numerator and denominator by the same number Look at examples on page 196

28. Writing Factions in Simplest Form Simplest form = when the numerator and denominator have no common factors except 1 Look at examples on page 197

29. 4-6 Rational Numbers Look at diagram on p. 205 Rational number = any number you can write as a fraction, with denominator NOT being a zero All integers are rational number because they can be written as a fraction Example: 5 can be written as 5/1

30. Writing Equivalent Fractions with Rational Numbers 1/2 -(4/5) 5/8 -(12/27)

31. Writing Equivalent Fractions with Rational Numbers - Answers 1/2 2/4, 4/8, 12/24 -(4/5)-(16/20), -(8/10) 5/810/16, 150/240 -(12/27)-(24/54), -(36/81)

32. Evaluate each expression 1. b + aa=(-2) b=(-3) 3a 2.b + 7 2a

33. Evaluate each expression - Answers 1. b + aa=(-2) b=(-3) 3a (-5)/(-6) or 5/6 2.b + 7 2a4/(-4) or -4/4 = -1

34. Graphing a Rational Number Graph each rational number on a number line Examples: 1/2 -(8/10) -0.2

35. 4-9 Scientific Notation Scientific Notation = is a way to write numbers using powers of 10 It lets you know the size of a number without having to count zeros Example: 7,500,000,000,000 7.5 x 10 12 (The first number must be greater than 1 but less than 10. The second number is a power of 10.)

36. Writing in Scientific Notation 0.000079 = 7.9 x 10 -5 89,000 = 8.9 x 10 4 Examples: 0.00000005 54,500,000

37. Writing in Scientific Notation - Answers 0.000079 = 7.9 x 10 -5 89,000 = 8.9 x 10 4 Examples: 0.00000005 = 5.0 x 10 -8 54,500,000 = 5.45 x 10 7

38. Writing in Standard Notation 8.9 x 10 5 = 890,000 2.71 x 10 -6 = 0.00000271 Examples: 3.21 x 10 7 5.9 x 10 -8

39. Writing in Standard Notation - Answers 8.9 x 10 5 = 890,000 2.71 x 10 -6 = 0.00000271 Examples: 3.21 x 10 7 = 32,100,000 5.9 x 10 -8 = 0.000000059

40. Multiplying with Scientific Notation (2.3 x 10 6 )(5 x 10 3 ) = (2.3 x 5) x (10 6 x 10 3) = 11.5 x (10 6 x 10 3) = 11.5 x 10 9 = 1.15 x 10 10 Examples:(5 x 10 6 )(6 x 10 2 ) (9 x 10 -3 )(7 x 10 8 ) (4.3 x 10 3 )(2 x 10 -8 )

41. Multiplying with Scientific Notation - Answers (2.3 x 10 6 )(5 x 10 3 ) = 2.3 x 5 x 10 6 x 10 3 = 11.5 x 10 6 x 10 3 = 11.5 x 10 9 = 1.15 x 10 10 Examples:(5 x 10 6 )(6 x 10 2 ) = 3.0 x 10 9 (9 x 10 -3 )(7 x 10 8 ) = 6.3 x 10 6 (4.3 x 10 3 )(2 x 10 -8 ) = 8.6 x 10 -5