Working with Square Roots. Return to Table of Contents

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1 Working with Square Roots Return to Table of Contents 36

2 Square Roots Recall... * Teacher Notes 37

3 Square Roots All of these numbers can be written with a square. Since the square is the inverse of the square root, they "undo" each other. 38

4 17 What is? 39

5 18 Find: 40

6 19 What is? 41

7 20 What is? 42

8 21 Find: 43

9 Variables What happens when you have variables in the radicand? To take the square root of a variable rewrite its exponent as the square of a power. 44

10 Variables IMPORTANT: When taking the square root of variables, remember that answers must be positive. Even powered answers, like the last page, will be positive even if the variables are negative. The same cannot be said if the answer has an odd power. When you take a square root and the answer has an odd power, put the result inside an absolute value symbol. 45

11 22 Simplify: A B C D 46

12 23 Simplify: A B C D 47

13 24 Simplify: A B C D 48

14 25 Simplify: A B C D 49

15 Square Roots of Fractions For square roots of fractions, take the square root the numerator (top) and denominator (bottom) separately. Teacher Notes 50

16 26 A B C D no real solution 51

17 27 A B C D no real solution 52

18 28 A B C D no real solution 53

19 29 A B C D no real solution 54

20 30 A B C D no real solution 55

21 Irrational Roots Return to Table of Contents 56

22 Simplifying Radicals is said to be a rational number because there is a perfect square that equals the radicand. If a radicand cannot be made into a perfect square, the root is said to be irrational, like. 57

23 Simplifying Radicals The commonly accepted form of a radical is called simplest radical form. To simplify numbers that are not perfect squares, start by breaking the radicand into factors and then breaking the factors into factors and so on until only prime numbers are left. This is called prime factorization. 58

24 Prime Factorization Examples of Prime Factorization: Note: There is maybe more than one way to break a part a number, but the prime factorization will always be the same. 59

25 31 Which of the following is the prime factorization of 24? A 3(8) B 4(6) C 2(2)(2)(3) D 2(2)(2)(3)(3) 60

26 32 Which of the following is the prime factorization of 72? A 9(8) B 2(2)(2)(2)(6) C 2(2)(2)(3) D 2(2)(2)(3)(3) 61

27 33 Which of the following is the prime factorization of 12? A 3(4) B 2(6) C 2(2)(2)(3) D 2(2)(3) 62

28 34 Which of the following is the prime factorization of 24 rewritten as powers of factors? A B C D 63

29 35 Which of the following is the prime factorization of 72 rewritten as powers of factors? A B C D 64

30 Simplifying Non-Perfect Square Radicands Find the prime factorization of the radicand, group prime factors to make perfect squares. Simplify. This is simplest radical form. Teacher Notes 65

31 36 Simplify: A B C D already in simplified form 66

32 37 Put in simplest radical form: A B C D already in simplified form 67

33 38 Put in simplest radical form: A B C D already in simplified form 68

34 39 Simplify: A B C D already in simplified form 69

35 40 Which of the following is not an irrational number? A B C D 70

36 Simplifying Radicals If there is a number, or expression, on the outside of the root remember that it is held together by multiplication. To simplify, put the root in simplest radical form and multiply. 45 Teacher Notes 71

37 41 Put in simplest radical form: A B C D 72

38 42 Simplify: A B C D 73

39 43 Put in simplest radical form: A B C D 74

40 44 Put in simplest radical form: A B C D 75

41 45 Put in simplest radical form: A B C D 76

42 Simplifying Radicals with Absolute Values The same process goes for variables, but absolute value signs need to be included where appropriate. Absolute value symbols are required when the initial exponent is even and the exponent after taking the root is odd. If the initial exponent is odd, you will not need absolute values. 77

43 Simplifying Radicals with Absolute Values Examples: Teacher Notes 78

44 46 Simplify: A B C D 79

45 47 Put in simplest radical form: A B C D 80

46 48 Simplify: A B C D 81

47 49 Put in simplest radical form: A B C D 82

48 50 Put in simplest radical form: A B C D 83

49 Cube Roots Return to Table of Contents 138

50 Cube Roots If a square root cancels a square, what cancels a cube? Teacher Notes 139

51 Cube Roots is read "the cube root of 64" A cube root looks like a square root. The difference is that little 3. The 3 (or other root) is called the index of a radical. Square roots have an index of two, but it is not usually written. An index indicates how many of each number or variable would allow taking a perfect root. A cube root is looking for groups of three, just as a square root is looking for groups of two. 140

52 Cube Roots The cube root and the cube cancel each other out. Examples: Teacher Notes 141

53 Try... Cube Roots Teacher Notes 142

54 Cube Roots Notice Where as is not real. Roots of negative numbers can only be taken if the index is odd. 143

55 82 Select all of the possible radicals that have real answers. A B C D E F G H I J 144

56 83 Evaluate the radical: A 3 B 4 C 6 D 8 145

57 84 Evaluate the radical: A 3 B 4 C 6 D 8 146

58 85 Evaluate the radical: A 2 B -2 C -4 D No real answer 147

59 86 Evaluate the radical: A -1 B - 1 / 3 C 1 /3 D 1 148

60 Cube Roots Just like square roots, cube roots can also be put in simplest radical form. Instead of looking for groups of 2, just look for groups of 3! Teacher Notes 149

61 Cube Roots The techniques and methods for solving square roots of variables, fractions, and decimals also work with cube roots. No absolute value signs are needed when using an odd index. Therefore, the answers for cube roots will not require absolute values. Teacher Notes 150

62 Cube Roots Put in simplest radical form: Teacher Notes 151

63 87 Simplify: A B C D not possible 152

64 88 Simplify: A B C D not possible 153

65 89 Simplify: A B C D not possible 154

66 90 Simplify: A B C D not possible 155

67 91 Simplify: A B C D not possible 156

68 92 Put in simplest radical form: A B C D not possible 157

69 n th Roots Return to Table of Contents 158

70 Absolute Values In general,. Absolute value signs are necessary if n is even, the initial exponent is even and the variable has an odd powered exponent after taking the root. Teacher Notes 159

71 n th Roots Try... Teacher Notes 160

72 93 Simplify: A B C D 161

73 94 Simplify: A B C D 162

74 95 Simplify: A B C D 163

75 96 Simplify: A B C D 164

76 97 Simplify: A B C D 165

77 98 Simplify: A B C D 166

78 99 Simplify: A B C D 167

79 100 Simplify: A B C D 168

80 Simplest Radical Form of Variables Divide the index into the exponent. The number of times the index goes into the exponent becomes the power on the outside of the radical and the remainder is the power of the radicand. 169

81 Absolute Value Signs As always, what about absolute value signs? An absolute value sign is needed if the index is even, the starting power of the variable is even and the answer outside the radical is an odd power. Examples of when absolute values are needed: 170

82 n th Roots Try... Teacher Notes 171

83 101 Simplify: A B C D 172

84 102 Simplify: A B C D 173

85 103 Simplify: A B C D 174

86 104 Simplify: A B C D 175

87 105 Simplify: A B C D 176

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