8th Grade Equations with Roots and Radicals

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1 Slide 1 / 87

2 Slide 2 / 87 8th Grade Equations with Roots and Radicals

3 Slide 3 / 87 Table of Contents Radical Expressions Containing Variables Click on topic to go to that section. Simplifying Non-Perfect Square Radicands Simplifying Roots of Variables Solving Equations with Perfect Square & Cube Roots Glossary & Standards

4 Slide 4 / 87 Radical Expressions Containing Variables Return to Table of Contents

5 Slide 5 / 87 Square Roots of Variables To take the square root of a variable rewrite its exponent as the square of a power. = (x 12 ) 2 = x 12 = (a 8 ) 2 = a 8 Can you find a shortcut to solve this type of problem? How would your shortcut make the problem easier?

6 Slide 5 (Answer) / 87 Square Roots of Variables To take the square root of a variable rewrite its exponent as the square of a power. = = Answer & Math Practice (x 12 ) 2 = x 12 (a 8 ) 2 = a 8 Answer: Divide the exponent inside of the square root by 2. The questions on this page address MP.8. [This object is a pull tab] Can you find a shortcut to solve this type of problem? How would your shortcut make the problem easier?

7 Slide 6 / 87 Square Roots of Variables If the square root of a variable raised to an even power has a variable raised to an odd power for an answer, the answer must have absolute value signs. This ensures that the answer will be positive. By Definition...

8 Slide 7 / 87 Square Root Practice Examples

9 Slide 8 / 87 Square Root Practice Try These. = x 5 = x 13

10 Slide 9 / 87 Square Root Practice How many of these expressions will need an absolute value sign when simplified? yes yes no no yes yes

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19 Slide 14 / 87 5 A B C D no real solution

20 Slide 14 (Answer) / 87 5 A B C Answer D no real solution C [This object is a pull tab]

21 Slide 15 / 87 Simplifying Non-Perfect Square Radicands Return to Table of Contents

22 Slide 16 / 87 Simplifying Perfect Squares (Review) A number is a perfect square if you can take that quantity of 1x1 unit squares and form them into a square. 1 1 Unit Square 4 is a perfect square, because you can take 4 unit squares and form them into a 2x2 square. (Notice that the square root of 4 is the length of one of its sides, since that side times itself equals 4.) = 2

23 Slide 17 / 87 Non-Perfect Squares What About Numbers that are not Perfect Squares? How can we simplify 8? 8 is not a perfect square, and no matter how we arrange the square units, we will not be able to form them into a square. So, we know that we will not have a whole number, which we can multiply by itself, to equal 8.

24 Slide 17 (Answer) / 87 Non-Perfect Squares Math Practice This What slide About and the Numbers next 5 slides that address are not Perfect Squares? MP.4: Model with mathematics MP.5: Use appropriate tools strategically by showing different How methods can of we simplifying simplify 8? square roots with visual aids, when applicable. When solving the example problems thereafter, Ask: What do you already know about this problem? (MP.4) Which tool/manipulative would be best for this problem? (MP.5) 8 Can is not you a do perfect this mentally? square, and (MP.5) no matter how we arrange the square Will a units, calculator we will help? not (MP.5) be able to form them into a square. What tools do you need? (MP.5) Why do the [This results object is make a pull tab] sense? (MP.4) So, we know that we will not have a whole number, which we can multiply by itself, to equal 8.

25 Slide 18 / 87 Non-Perfect Squares What happens when the radicand is not a perfect square? 8 Rewrite the radicand as a product of its largest perfect square factor. click 8 = Simplify the square root of the perfect square. click When simplified form still contains a radical, it is said to be irrational.

26 Slide 19 / 87 Non-Perfect Squares What happens when the radicand is not a perfect square? 1. Rewrite the radicand as a product of its largest perfect square factor. 2. Simplify the square root of the perfect square. click click click When simplified form still contains a radical, it is said to be irrational.

27 Slide 20 / 87 Simplifying Non-Perfect Squares Identifying the largest perfect square factor when simplifying radicals will result in the least amount of work. Ex: Not simplified! Keep going! Finding the largest perfect square factor results in less work: Note that the answers are the same for both solution processes

28 Slide 21 / 87 Simplifying Non-Perfect Squares Another method for simplifying non-perfect squares is to use prime factorization and a factor tree. For example, 48 can be broken down as follows:

29 Slide 22 / 87 Simplifying Non-Perfect Squares (2) 3 = After you factor the number into all of its primes, you can circle each pair of numbers that exist to signify that they come outside of the radical. For each pair circled, one number comes out. If more than one pair of numbers are circled, join the numbers outside of the radical by a multiplication sign. Any numbers left without a match must stay inside of the radical. Multiply them together, if needed. Therefore, 48 simplifies to 4 3.

30 Slide 22 (Answer) / 87 Simplifying Non-Perfect Squares Teacher Notes You can add 48 a storyline for this method. For example, if the factors of 48 attend a speed dating 2 24party, each prime factor is looking for its match. If the prime 2(2) factors 3 = 4 3 find their match, 2 12 they walk out as one couple. If any factors can't find their match, they must 2 6remain at the party. Another one could 2 3be to "get out of jail", each prime number needs a "buddy" to After escape. you factor the number into all of its primes, you can circle each pair of numbers that exist to signify that they come outside of [This object is a pull tab] the radical. For each pair circled, one number comes out. If more than one pair of numbers are circled, join the numbers outside of the radical by a multiplication sign. Any numbers left without a match must stay inside of the radical. Multiply them together, if needed. Therefore, 48 simplifies to 4 3.

31 Slide 23 / 87 Try These. Non-Perfect Squares Practice

32 Try These. Prime Factoring Answer Slide 23 (Answer) / 87 Non-Perfect Squares Practice (3) (3) 2(5) [This object is a pull tab]

33 Slide 24 / 87 6 Simplify A B C D already in simplified form

34 Slide 24 (Answer) / 87 6 Simplify A B C D Answer already in simplified form A [This object is a pull tab]

35 Slide 25 / 87 7 Simplify A B C D already in simplified form

36 Slide 25 (Answer) / 87 7 Simplify A B C D Answer already in simplified form B [This object is a pull tab]

37 Slide 26 / 87 8 Simplify A B C D already in simplified form

38 Slide 26 (Answer) / 87 8 Simplify A B C D Answer already in simplified form A [This object is a pull tab]

39 Slide 27 / 87 9 Simplify A B C D already in simplified form

40 Slide 27 (Answer) / 87 9 Simplify A B C D Answer already in simplified form D [This object is a pull tab]

41 Slide 28 / Simplify A B C D already in simplified form

42 Slide 28 (Answer) / Simplify A B C Answer D already in simplified form B [This object is a pull tab]

43 Slide 29 / Simplify A B C D already in simplified form

44 Slide 29 (Answer) / Simplify A B C D Answer already in simplified form B [This object is a pull tab]

45 Slide 30 / Which of the following does not have an irrational simplified form? A B C D

46 Slide 30 (Answer) / Which of the following does not have an irrational simplified form? A B C D Answer D [This object is a pull tab]

47 Slide 31 / The diagonal of a square can be expressed by the formula d= 2a 2, where a is the side length of the square. Select the correct options to show the length of the diagonal of the square shown. Your answer should be a radicand in simplest form. d = A 3 B 4 C 9 D 1 E 2 F 3 9

48 Slide 31 (Answer) / The diagonal of a square can be expressed by the formula d= 2a 2, where a is the side length of the square. Select the correct options to show the length of the diagonal of the square shown. Your answer should be a radicand in simplest form. d = Answer C, E 9 A 3 B 4 C 9 D 1 E 2 F 3 [This object is a pull tab]

49 Slide 32 / The distance, d, in miles that a person can see to the horizon is calculated with the following formula. d = 3h 2 h = the person's height above sea level in feet. How far to the horizon would you be able to see from this vantage point? Your answer should be a radicand in simplest form. 100 ft above sea level d = A 3 B 4 C 5 D 5 E 6 F 10

50 Slide 32 (Answer) / The distance, d, in miles that a person can see to the horizon is calculated with the following formula. d = 3h 2 h = the person's height above 300 sea is level divisible in feet. by 2. How far to the horizon would you be able to see from this vantage point? Your answer should be a radicand in simplest form. Answer So, 100 ft above sea level = 150 d = A 3 B 4 C 5 D 5 E 6 F 10 C, E [This object is a pull tab]

51 Slide 33 / 87 Simplest Radical Form Note - If a radical begins with a coefficient before the radicand is simplified, any perfect square that is simplified will be multiplied by the existing coefficient. (multiply the outside) 2

52 Slide 34 / 87 Simplest Radical Form Likewise - If a radical begins with a coefficient before the radicand is simplified, any pair of primes that are circled will be multiplied by the existing coefficient. (multiply the outside) (3) (2)

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55 Slide 36 / Simplify A B C D

56 Slide 36 (Answer) / Simplify A B C D Answer A [This object is a pull tab]

57 Slide 37 / Simplify A B C D

58 Slide 37 (Answer) / Simplify A B C D Answer B [This object is a pull tab]

59 Slide 38 / Simplify A B C D

60 Slide 38 (Answer) / Simplify A B C D Answer B [This object is a pull tab]

61 Slide 39 / Simplify A B C D

62 Slide 39 (Answer) / Simplify A B C D Answer A [This object is a pull tab]

63 Slide 40 / Simplify A B C D

64 Slide 40 (Answer) / Simplify A B C D Answer C [This object is a pull tab]

65 Slide 41 / 87 Teachers: Use the questions found in the pull tab for the next 2 slides.

66 Slide 41 (Answer) / 87 MP.1: Make sense of problems and persevere in solving them. Teachers: MP.2: Reasoning quantitatively and abstractly. Use the questions found in the pull tab for the next 2 Ask: slides. What facts do you have? (MP.1 & MP.2) How could you start this problem? (MP.1) What does the letter/number _ represent in the problem? (MP.2) Math Practice [This object is a pull tab]

67 Slide 42 / When is written in simplest radical form, the result is. What is the value of k? A 20 B 10 C 7 D 4 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

68 Slide 42 (Answer) / When is written in simplest radical form, the result is. What is the value of k? A 20 B 10 C 7 D 4 Answer B [This object is a pull tab] From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

69 Slide 43 / When is expressed in simplest form, what is the value of a? A 6 B 2 C 3 D 8 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

70 Slide 43 (Answer) / When is expressed in simplest form, what is the value of a? A 6 B 2 C 3 D 8 Answer A [This object is a pull tab] From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

71 Slide 44 / Which is greater or 6? Derived from

72 Slide 44 (Answer) / Which is greater or 6? Answer 6 [This object is a pull tab] Derived from

73 Slide 45 / Which is greater or 10? Derived from

74 Slide 45 (Answer) / Which is greater or 10? Answer 10 [This object is a pull tab] Derived from

75 Slide 46 / 87 Simplifying Roots of Variables Return to Table of Contents

76 Slide 47 / 87 Using Absolute Value When we simplify radicals, we are told to assume all variables are positive. But, why? Because, the square root of the square of a negative number is not the original number.

77 Slide 48 / 87 Using Absolute Value Take -2 for example. (-2) 2 = +4 But, 4 is not -2, it is +2. By definition square roots of numbers are positive. You started with a negative number (-2), and ended up with a positive number (+2). So, the square root of a number is the absolute value of the square root. 4 = 2 This accounts for +2 2 and (-2) 2.

78 Slide 48 (Answer) / 87 Using Absolute Value Take -2 for example. (-2) 2 = +4 MP.6: Attend to precision Emphasize the use of parentheses when But, 4 is not -2, raising it is +2. any negative number to a power. It shows how the negative sign is included each time the multiplication By definition square roots of numbers are positive. takes place. For example: You started with a (-2) negative 2 = (-2)(-2) number = 4 (-2), and ended up with a positive andnumber (+2) = -(2)(2) = -4 So, the square root of a number is the absolute value of the square root. [This object is a pull tab] Math Practice 4 = 2 This accounts for +2 2 and (-2) 2.

79 Slide 49 / 87 Using Absolute Value Easy enough. But what about when the radicand is a variable, and we don't know the sign of the unknown value? x 2 Is x positive or negative? We can't know, so we "assume all variables are positive".

80 Slide 50 / 87 Simplifying Roots of Variables The technical definition of "the square root of x squared" is "the absolute value of x". x 2 = x x x = x 2 x is positive x x - - = x 2 x is negative

81 Slide 51 / 87 Simplifying Roots of Variables Using Absolute Values When working with square roots, an absolute value sign is needed if: The power of the given variable is even. and The answer contains a variable raised to an odd power outside the radical. x 6 x 3 x 6 = x 3

82 Slide 52 / 87 But, Why? x 6 = x 3 x x x x x x = x x x Whether x is positive or negative, when it is multiplied by itself an even number of times, it will turn out to be a positive number. So, x is positive. However, if x is negative, when it is multiplied by itself an odd number of times, it will turn out to be a negative number. So, x could be negative. So, in order for x 6 = x 3, we must use an absolute value sign to indicate that x is positive. x 6 = x 3

83 Slide 53 / 87 Roots of Variable Practice More Examples Use expanded form to explain why absolute value must be used in these answers.

84 Slide 53 (Answer) / 87 Roots of Variable Practice More Examples Use expanded form to explain why absolute MP.3: Construct value viable must arguments be used in and these answers. critique the reasoning of others. MP.7: Look for and make use of structure. Math Practice Ask: Why do we need to use the absolute value in these problems? (MP.7) What do you know about taking square roots of numbers and the value of odd exponential terms that can apply to this problem? (MP.7) How can you prove that your answer is correct? (MP.3) [This object is a pull tab]

85 Slide 54 / 87 Simplifying Roots of Variables Divide the exponent by 2. The number of times that 2 goes into the exponent becomes the power on the outside of the radical and the remainder is the power of the radicand. x 7 = x x x x x x x = x 3 x Note: Absolute value signs are not needed because the radicand had an odd power to start.

86 Examples: Slide 55 / 87 Roots of Variables Examples Combining it all: 50x 4 y 12 z 3 z zz 25 2(x 2 ) 2 (y 6 ) 2 5 x 2 y 6 z 2z

87 Slide 56 / 87 Roots of Variables Practice Only the y has an odd power on the outside of the radical. The x had an odd power under the radical so no absolute value signs needed. The m's starting power was odd, so it does not require absolute value signs.

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92 Slide 59 / Simplify A B C D

93 Slide 59 (Answer) / Simplify A B C D Answer C [This object is a pull tab]

94 Slide 60 / Simplify A B C D

95 Slide 60 (Answer) / Simplify A B C D Answer A [This object is a pull tab]

96 Slide 61 / 87 Solving Equations with Perfect Square and Cube Roots Return to Table of Contents

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98 Slide 63 / 87 Squares and Cubes Practice Use the numbers shown to make the equations true. Each number can be used only once. (Problem from ) a. = b. 3 =

99 Slide 63 (Answer) / 87 Squares and Cubes Practice Use the numbers shown to make the equations true. Each number can be used only once. (Problem from ) Answer a. = b. 3 = [This object is a pull tab]

100 Slide 64 / 87 Squares and Cubes Practice Complete the Venn-Diagram to classify the numbers as perfect squares and perfect cubes (Problem from ) Perfect Squares Perfect Cubes

101 Slide 64 (Answer) / 87 Squares and Cubes Practice Complete the Venn-Diagram to classify the numbers as perfect squares and perfect cubes (Problem from ) Answer [This object is a pull tab] Perfect Squares Perfect Cubes

102 Slide 65 / 87 Solving Equations When we solve equations, the solution sometimes requires finding a square or cube root of both sides of the equation. When your equation simplifies to: x 2 = # you must find the square root of both sides in order to find the value of x. When your equation simplifies to: x 3 = # you must find the cube root of both sides in order to find the value of x.

103 Slide 66 / 87 Solving Equations Example Example: Solve. = Divide each side by the coefficient. Then take the square root of each side.

104 Slide 67 / 87 Example: Solving Equations Example Solve. Multiply each side by nine, then take the cube root of each side.

105 Slide 68 / 87 Notice! The answer is only a positive 3, not Why is the answer only positive and not both positive and negative?

106 Slide 69 / 87 Cube Roots The cube root of 27 is 3, and not -3, because when 3 is cubed you get x 3 x 3 = 27 If you were to cube -3, you would get x -3 x -3 = -27 Therefore, the cube root of -27 is -3. So we can take a cube root of a positive number AND take the cube root of a negative number!

107 Slide 70 / 87 Cube Roots Examples

108 Try These: Solve. Slide 71 / 87 Squares and Cubes Practice ± 10 ± 8 ± 9 ± 7

109 Try These: Solve. Slide 72 / 87 Squares and Cubes Practice

110 28 Solve. Slide 73 / 87

111 Slide 73 (Answer) / Solve. Answer ±12 [This object is a pull tab]

112 29 Solve. Slide 74 / 87

113 Slide 74 (Answer) / Solve. Answer ±12 [This object is a pull tab]

114 30 Solve. Slide 75 / 87

115 Slide 75 (Answer) / Solve. Answer 2 [This object is a pull tab]

116 31 Solve. Slide 76 / 87

117 Slide 76 (Answer) / Solve. Answer 4 [This object is a pull tab]

118 Slide 77 / Solve 15 + x 2 = 40 Derived from

119 Slide 77 (Answer) / Solve 15 + x 2 = 40 Answer ±5 [This object is a pull tab] Derived from

120 Slide 78 / Solve 2 + x 3 = 10 Derived from

121 Slide 78 (Answer) / Solve 2 + x 3 = 10 Answer 2 [This object is a pull tab] Derived from

122 Slide 79 / A cube has a volume of 343 cm 3. a) Write an equation that could be used to determine the length, L, of one side. b) Solve the equation. Derived from

123 Slide 79 (Answer) / A cube has a volume of 343 cm 3. a) Write an equation that could be used to determine the length, L, of one side. b) Solve the equation. Answer a) L 3 = 343 b) L = 7 cm [This object is a pull tab] Derived from

124 Slide 80 / Estimate the area of the rectangle to the nearest tenth.

125 Slide 80 (Answer) / Estimate the area of the rectangle to the nearest tenth. Answer u 2 [This object is a pull tab]

126 Slide 81 / If the area of a square is square inches, what is the length, in inches, of one side of the square? A B C D

127 Slide 81 (Answer) / If the area of a square is square inches, what is the length, in inches, of one side of the square? A B C D Answer B [This object is a pull tab]

128 Slide 82 / Which equation has both 4 and -4 as possible values of y? A B C D From PARCC EOY sample test non-calculator #9

129 Slide 82 (Answer) / Which equation has both 4 and -4 as possible values of y? A B C Answer C D [This object is a pull tab] From PARCC EOY sample test non-calculator #9

130 Slide 83 / 87 Glossary & Standards Return to Table of Contents

131 Slide 84 / 87 Cube To multiply a number by itself and then again by itself. The product of three equal factors. What is 4 cubed? 4 3 = 4 x 4 x 4 = (4)(4)(4) = 64 What is the cube of 6? 6 3 = 6 x 6 x 6 = (6)(6)(6) = 216 What is 10 cubed? 10 3 = 10 x 10 x 10 = (10)(10)(10) = 1000 Back to Instruction

132 Slide 85 / 87 Cube Root A value that, when used in a multiplication three times, gives that number. Symbol: 3 "cube root" 3 64 = 4 (4)(4)(4) = 64 4x4x4 = = 6 (6)(6)(6) = 216 6x6x6 = 216 Back to Instruction

133 Slide 86 / 87 Power A power is another name for an exponent. It is a small, raised number that shows how many times to multiply the base by itself. Power 3 2 Base "3 to the second power" 3 2 = 3x = 3 x 3 x x x 3 3 Back to Instruction

134 Slide 87 / 87 Standards for Mathematical Practice MP1 Making sense of problems & persevere in solving them. MP2 Reason abstractly & quantitatively. MP3 Construct viable arguments and critique the reasoning of others. MP4 Model with mathematics. MP5 Use appropriate tools strategically. MP6 Attend to precision. MP7 Look for & make use of structure. MP8 Look for & express regularity in repeated reasoning. Click on each standard to bring you to an example of how to meet this standard within the unit.