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7.5 – Radical Expressions. Radical Radical – radical symbol.

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Presentation on theme: "7.5 – Radical Expressions. Radical Radical – radical symbol."— Presentation transcript:

1 7.5 – Radical Expressions

2 Radical

3 Radical – radical symbol

4 Radical – radical symbol =

5 Recall:

6 Radical – radical symbol = Recall: Ex. 1 Simplify

7 Radical – radical symbol = Recall: Ex. 1 Simplify = ____ √ 3·3·3

8 Radical – radical symbol = Recall: Ex. 1 Simplify = 2 ____ √ 3·3·3

9 Radical – radical symbol = Recall: Ex. 1 Simplify = 2 ____ √ 3·3·3

10 Radical – radical symbol = Recall: Ex. 1 Simplify = 2 √ 3·3·3 √ 3·3 · √ 3

11 Radical – radical symbol = Recall: Ex. 1 Simplify = 2 √ 3·3·3 √ 3·3 · √ 3 3 · √ 3

12 Radical – radical symbol = Recall: Ex. 1 Simplify = 2 √ 3·3·3 √ 3·3 · √ 3 3 √ 3

13 Ex. 2 Simplify √ 16p 8 q 7

14 √ 16p 8 q 7 =

15 Ex. 2 Simplify √ 16p 8 q 7 √ 16p 8 q 7 = √ 16

16 Ex. 2 Simplify √ 16p 8 q 7 √ 16p 8 q 7 = √ 16 · √ p 8

17 Ex. 2 Simplify √ 16p 8 q 7 √ 16p 8 q 7 = √ 16 · √ p 8 · √ q 7

18 Ex. 2 Simplify √ 16p 8 q 7 √ 16p 8 q 7 = √ 16 · √ p 8 · √ q 7 = √4 2

19 Ex. 2 Simplify √ 16p 8 q 7 √ 16p 8 q 7 = √ 16 · √ p 8 · √ q 7 = √4 2 ·√(p 4 ) 2

20 Ex. 2 Simplify √ 16p 8 q 7 √ 16p 8 q 7 = √ 16 · √ p 8 · √ q 7 = √4 2 ·√(p 4 ) 2 ·√(q 6 )q

21 Ex. 2 Simplify √ 16p 8 q 7 √ 16p 8 q 7 = √ 16 · √ p 8 · √ q 7 = √4 2 ·√(p 4 ) 2 ·√(q 6 )q = √4 2 ·√(p 4 ) 2 ·√(q 3 ) 2 ·q

22 Ex. 2 Simplify √ 16p 8 q 7 √ 16p 8 q 7 = √ 16 · √ p 8 · √ q 7 = √4 2 ·√(p 4 ) 2 ·√(q 6 )q = √4 2 ·√(p 4 ) 2 ·√(q 3 ) 2 ·q = √4 2 ·√(p 4 ) 2 ·√(q 3 ) 2 ·√q

23 Ex. 2 Simplify √ 16p 8 q 7 √ 16p 8 q 7 = √ 16 · √ p 8 · √ q 7 = √4 2 ·√(p 4 ) 2 ·√(q 6 )q = √4 2 ·√(p 4 ) 2 ·√(q 3 ) 2 ·q = √4 2 ·√(p 4 ) 2 ·√(q 3 ) 2 ·√q = 4 · p 4 · q 3 ·√q

24 Ex. 2 Simplify √ 16p 8 q 7 √ 16p 8 q 7 = √ 16 · √ p 8 · √ q 7 = √4 2 ·√(p 4 ) 2 ·√(q 6 )q = √4 2 ·√(p 4 ) 2 ·√(q 3 ) 2 ·q = √4 2 ·√(p 4 ) 2 ·√(q 3 ) 2 ·√q = 4 · p 4 · q 3 ·√q = 4p 4 q 3 √q

25 Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 33

26 Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 33

27 Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 33

28 Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 33

29 Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 · √ 9n 2 · 24n 33 3

30 Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 · √ 9n 2 · 24n 18 · √ 3 2 n 2 · 3·8·n 33 3 3

31 Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 · √ 9n 2 · 24n 18 · √ 3 2 n 2 · 3·8·n 33 3 3

32 Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 · √ 9n 2 · 24n 18 · √ 3 2 n 2 · 3·8·n 33 3 3

33 Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 · √ 9n 2 · 24n 18 · √ 3 2 n 2 · 3·8·n 18 · √ 3 2 ·n 2 ·3·2 3 ·n 33 3 3 3

34 Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 · √ 9n 2 · 24n 18 · √ 3 2 n 2 · 3·8·n 18 · √ 3 2 ·n 2 ·3·2 3 ·n 33 3 3 3

35 Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 · √ 9n 2 · 24n 18 · √ 3 2 n 2 · 3·8·n 18 · √ 3 2 ·n 2 ·3·2 3 ·n 18 · √ 3 3 ·n 3 ·2 3 33 3 3 3 3

36 Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 · √ 9n 2 · 24n 18 · √ 3 2 n 2 · 3·8·n 18 · √ 3 2 ·n 2 ·3·2 3 ·n 18 · √ 3 3 ·n 3 ·2 3 33 3 3 3 3

37 Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 · √ 9n 2 · 24n 18 · √ 3 2 n 2 · 3·8·n 18 · √ 3 2 ·n 2 ·3·2 3 ·n 18 · √ 3 3 ·n 3 ·2 3 33 3 3 3 3

38 Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 · √ 9n 2 · 24n 18 · √ 3 2 n 2 · 3·8·n 18 · √ 3 2 ·n 2 ·3·2 3 ·n 18 · √ 3 3 ·n 3 ·2 3 18 · √ 3 3 ·√ n 3 ·√ 2 3 33 3 3 3 3 333

39 Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 · √ 9n 2 · 24n 18 · √ 3 2 n 2 · 3·8·n 18 · √ 3 2 ·n 2 ·3·2 3 ·n 18 · √ 3 3 ·n 3 ·2 3 18 · √ 3 3 ·√ n 3 ·√ 2 3 18 · 3 · n · 2 33 3 3 3 3 333

40 Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 · √ 9n 2 · 24n 18 · √ 3 2 n 2 · 3·8·n 18 · √ 3 2 ·n 2 ·3·2 3 ·n 18 · √ 3 3 ·n 3 ·2 3 18 · √ 3 3 ·√ n 3 ·√ 2 3 18 · 3 · n · 2 18 · 3 · 2 · n 33 3 3 3 3 333

41 Ex. 3 Simplify each expression. 6√ 9n 2 · 3√ 24n 6 · 3 · √ 9n 2 · 24n 18 · √ 3 2 n 2 · 3·8·n 18 · √ 3 2 ·n 2 ·3·2 3 ·n 18 · √ 3 3 ·n 3 ·2 3 18 · √ 3 3 ·√ n 3 ·√ 2 3 18 · 3 · n · 2 18 · 3 · 2 · n 108n 33 3 3 3 3 333

42 Ex. 4 Simplify each expression. a. 2√ 12 - 3√ 27 + 2√ 48

43 Ex. 4 Simplify each expression. a. 2√ 12 - 3√ 27 + 2√ 48 2√ 2·2·3 - 3√ 3·3·3 + 2√ 2·2·2·2·3

44 Ex. 4 Simplify each expression. a. 2√ 12 - 3√ 27 + 2√ 48 2√ 2·2·3 - 3√ 3·3·3 + 2√ 2·2·2·2·3 2√ 2 2 ·3 - 3√ 3 3 + 2√ 2 4 ·3

45 Ex. 4 Simplify each expression. a. 2√ 12 - 3√ 27 + 2√ 48 2√ 2·2·3 - 3√ 3·3·3 + 2√ 2·2·2·2·3 2√ 2 2 ·3 - 3√ 3 3 + 2√ 2 4 ·3 2√ 2 2 ·3 - 3√ 3 3 + 2√ 2 2 ·2 2 ·3

46 Ex. 4 Simplify each expression. a. 2√ 12 - 3√ 27 + 2√ 48 2√ 2·2·3 - 3√ 3·3·3 + 2√ 2·2·2·2·3 2√ 2 2 ·3 - 3√ 3 3 + 2√ 2 4 ·3 2√ 2 2 ·3 - 3√ 3 3 + 2√ 2 2 ·2 2 ·3 2·2·√3 - 3·3·√3 + 2 ·2·2· √3

47 Ex. 4 Simplify each expression. a. 2√ 12 - 3√ 27 + 2√ 48 2√ 2·2·3 - 3√ 3·3·3 + 2√ 2·2·2·2·3 2√ 2 2 ·3 - 3√ 3 3 + 2√ 2 4 ·3 2√ 2 2 ·3 - 3√ 3 3 + 2√ 2 2 ·2 2 ·3 2·2·√3 - 3·3·√3 + 2 ·2·2· √3 4√ 3 - 9√ 3 + 8√ 3

48 Ex. 4 Simplify each expression. a. 2√ 12 - 3√ 27 + 2√ 48 2√ 2·2·3 - 3√ 3·3·3 + 2√ 2·2·2·2·3 2√ 2 2 ·3 - 3√ 3 3 + 2√ 2 4 ·3 2√ 2 2 ·3 - 3√ 3 3 + 2√ 2 2 ·2 2 ·3 2·2·√3 - 3·3·√3 + 2 ·2·2· √3 4√ 3 - 9√ 3 + 8√ 3

49 Ex. 4 Simplify each expression. a. 2√ 12 - 3√ 27 + 2√ 48 2√ 2·2·3 - 3√ 3·3·3 + 2√ 2·2·2·2·3 2√ 2 2 ·3 - 3√ 3 3 + 2√ 2 4 ·3 2√ 2 2 ·3 - 3√ 3 3 + 2√ 2 2 ·2 2 ·3 2·2·√3 - 3·3·√3 + 2 ·2·2· √3 4√ 3 - 9√ 3 + 8√ 3 -5√ 3 + 8√ 3

50 Ex. 4 Simplify each expression. a. 2√ 12 - 3√ 27 + 2√ 48 2√ 2·2·3 - 3√ 3·3·3 + 2√ 2·2·2·2·3 2√ 2 2 ·3 - 3√ 3 3 + 2√ 2 4 ·3 2√ 2 2 ·3 - 3√ 3 3 + 2√ 2 2 ·2 2 ·3 2·2·√3 - 3·3·√3 + 2 ·2·2· √3 4√ 3 - 9√ 3 + 8√ 3 -5√ 3 + 8√ 3

51 Ex. 4 Simplify each expression. a. 2√ 12 - 3√ 27 + 2√ 48 2√ 2·2·3 - 3√ 3·3·3 + 2√ 2·2·2·2·3 2√ 2 2 ·3 - 3√ 3 3 + 2√ 2 4 ·3 2√ 2 2 ·3 - 3√ 3 3 + 2√ 2 2 ·2 2 ·3 2·2·√3 - 3·3·√3 + 2 ·2·2· √3 4√ 3 - 9√ 3 + 8√ 3 -5√ 3 + 8√ 3 3√ 3

52 b. (5√ 3 – 6)(5√ 3 +6)

53

54 5√ 3(5√ 3 +6)

55 b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6)

56 b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6)

57 b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6)

58 b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3

59 b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3

60 b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6

61 b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6

62 b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3

63 b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3

64 b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3 – 6·6

65 b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3 – 6·6 5·5√ 3·√ 3 + 5·6√ 3 – 6·5√ 3 – 6·6

66 b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3 – 6·6 5·5√ 3·√ 3 + 5·6√ 3 – 6·5√ 3 – 6·6

67 b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3 – 6·6 5·5√ 3·√ 3 + 5·6√ 3 – 6·5√ 3 – 6·6

68 b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3 – 6·6 5·5√ 3·√ 3 + 5·6√ 3 – 6·5√ 3 – 6·6

69 b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3 – 6·6 5·5√ 3·√ 3 + 5·6√ 3 – 6·5√ 3 – 6·6 25√ 9 + 30√ 3 – 30√ 3 – 36

70 b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3 – 6·6 5·5√ 3·√ 3 + 5·6√ 3 – 6·5√ 3 – 6·6 25√ 9 + 30√ 3 – 30√ 3 – 36

71 b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3 – 6·6 5·5√ 3·√ 3 + 5·6√ 3 – 6·5√ 3 – 6·6 25√ 9 + 30√ 3 – 30√ 3 – 36 25·3

72 b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3 – 6·6 5·5√ 3·√ 3 + 5·6√ 3 – 6·5√ 3 – 6·6 25√ 9 + 30√ 3 – 30√ 3 – 36 25·3

73 b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3 – 6·6 5·5√ 3·√ 3 + 5·6√ 3 – 6·5√ 3 – 6·6 25√ 9 + 30√ 3 – 30√ 3 – 36 25·3+ 0 – 36

74 b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3 – 6·6 5·5√ 3·√ 3 + 5·6√ 3 – 6·5√ 3 – 6·6 25√ 9 + 30√ 3 – 30√ 3 – 36 25·3 – 36 75 – 36

75 b. (5√ 3 – 6)(5√ 3 +6) 5√ 3(5√ 3 +6) – 6(5√ 3 +6) 5√ 3·5√ 3 + 5√ 3·6 – 6·5√ 3 – 6·6 5·5√ 3·√ 3 + 5·6√ 3 – 6·5√ 3 – 6·6 25√ 9 + 30√ 3 – 30√ 3 – 36 25·3 – 36 75 – 36 39

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