8th Grade. Slide 1 / 87. Slide 2 / 87. Slide 3 / 87. Equations with Roots and Radicals. Table of Contents

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1 Slide 1 / 87 Slide 2 / 87 8th Grade Equations with Roots and Radicals Table of ontents Slide 3 / 87 Radical Expressions ontaining Variables Simplifying Non-Perfect Square Radicands Simplifying Roots of Variables Solving Equations with Perfect Square & ube Roots Glossary & Standards lick on topic to go to that section.

2 Slide 4 / 87 Radical Expressions ontaining Variables Return to Table of ontents Square Roots of Variables Slide 5 / 87 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 an you find a shortcut to solve this type of problem? How would your shortcut make the problem easier? Square Roots of Variables Slide 5 (nswer) / 87 To take the square root of a variable rewrite its exponent as the square of a power. = = nswer & Math Practice (x 12 ) 2 = x 12 nswer: ivide the exponent inside of the square root by 2. The questions on this page address (a 8 ) 2 = a 8 MP.8. an you find a shortcut to solve this type of problem? How would your shortcut make the problem easier?

3 Square Roots of Variables Slide 6 / 87 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. y efinition... Square Root Practice Slide 7 / 87 Examples Square Root Practice Slide 8 / 87 Try These. = x 5 = x 13

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

5 Slide 11 / 87 Slide 11 (nswer) / 87 Slide 12 / 87

6 Slide 12 (nswer) / 87 Slide 13 / 87 Slide 13 (nswer) / 87

7 5 Slide 14 / 87 no real solution 5 Slide 14 (nswer) / 87 nswer no real solution Slide 15 / 87 Simplifying Non-Perfect Square Radicands Return to Table of ontents

8 Simplifying Perfect Squares (Review) Slide 16 / 87 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 4 = 2 2 Non-Perfect Squares Slide 17 / 87 What bout 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. Non-Perfect Squares Slide 17 (nswer) / 87 Math Practice This What slide bout 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, sk: What do you already know about this problem? (MP.4) Which tool/manipulative would be best for this problem? (MP.5) 8 an 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.

9 Non-Perfect Squares Slide 18 / 87 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. Non-Perfect Squares Slide 19 / 87 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. Simplifying Non-Perfect Squares Identifying the largest perfect square factor when simplifying radicals will result in the least amount of work. Slide 20 / 87 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

10 Simplifying Non-Perfect Squares Slide 21 / 87 nother method for simplifying non-perfect squares is to use prime factorization and a factor tree. For example, 48 can be broken down as follows: Simplifying Non-Perfect Squares Slide 22 / (2) 3 = fter 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. ny numbers left without a match must stay inside of the radical. Multiply them together, if needed. Therefore, 48 simplifies to 4 3. Simplifying Non-Perfect Squares Slide 22 (nswer) / 87 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. nother one could 2 3be to "get out of jail", each prime number needs a "buddy" to fter 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 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. ny numbers left without a match must stay inside of the radical. Multiply them together, if needed. Therefore, 48 simplifies to 4 3.

11 Try These. Non-Perfect Squares Practice Slide 23 / 87 Try These. Prime Factoring nswer Non-Perfect Squares Practice (3) (3) 2(5) Slide 23 (nswer) / 87 6 Simplify Slide 24 / 87 already in simplified form

12 6 Simplify Slide 24 (nswer) / 87 nswer already in simplified form 7 Simplify Slide 25 / 87 already in simplified form 7 Simplify Slide 25 (nswer) / 87 nswer already in simplified form

13 8 Simplify Slide 26 / 87 already in simplified form 8 Simplify Slide 26 (nswer) / 87 nswer already in simplified form 9 Simplify Slide 27 / 87 already in simplified form

14 9 Simplify Slide 27 (nswer) / 87 nswer already in simplified form 10 Simplify Slide 28 / 87 already in simplified form 10 Simplify Slide 28 (nswer) / 87 nswer already in simplified form

15 11 Simplify Slide 29 / 87 already in simplified form 11 Simplify Slide 29 (nswer) / 87 nswer already in simplified form 12 Which of the following does not have an irrational simplified form? Slide 30 / 87

16 12 Which of the following does not have an irrational simplified form? Slide 30 (nswer) / 87 nswer 13 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. Slide 31 / 87 d = E 2 F 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. Slide 31 (nswer) / 87 d = nswer, E E 2 F 3

17 14 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. Slide 32 / 87 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 = E 6 F The distance, d, in miles that a person can see to the horizon is calculated with the following formula. d = 3h 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. nswer h = the person's height above 300 sea is level divisible in feet. by 2. So, 100 ft above sea level = 150 Slide 32 (nswer) / 87 d = E 6 F 10, E Simplest Radical Form Slide 33 / 87 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

18 Simplest Radical Form Slide 34 / 87 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) Slide 35 / 87 Slide 35 (nswer) / 87

19 15 Simplify Slide 36 / Simplify Slide 36 (nswer) / 87 nswer 16 Simplify Slide 37 / 87

20 16 Simplify Slide 37 (nswer) / 87 nswer 17 Simplify Slide 38 / Simplify Slide 38 (nswer) / 87 nswer

21 18 Simplify Slide 39 / Simplify Slide 39 (nswer) / 87 nswer 19 Simplify Slide 40 / 87

22 19 Simplify Slide 40 (nswer) / 87 nswer Slide 41 / 87 Teachers: Use the questions found in the pull tab for the next 2 slides. Slide 41 (nswer) / 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 sk: 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

23 20 When is written in simplest radical form, the result is. What is the value of k? Slide 42 / From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from accessed 17, June, When is written in simplest radical form, the result is. What is the value of k? Slide 42 (nswer) / nswer From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from accessed 17, June, When is expressed in simplest form, what is the value of a? Slide 43 / From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from accessed 17, June, 2011.

24 21 When is expressed in simplest form, what is the value of a? Slide 43 (nswer) / nswer From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from accessed 17, June, Which is greater or 6? Slide 44 / 87 erived from 22 Which is greater or 6? Slide 44 (nswer) / 87 nswer 6 erived from

25 23 Which is greater or 10? Slide 45 / 87 erived from 23 Which is greater or 10? Slide 45 (nswer) / 87 nswer 10 erived from Slide 46 / 87 Simplifying Roots of Variables Return to Table of ontents

26 Using bsolute Value Slide 47 / 87 When we simplify radicals, we are told to assume all variables are positive. ut, why? ecause, the square root of the square of a negative number is not the original number. Using bsolute Value Slide 48 / 87 Take -2 for example. (-2) 2 = +4 ut, 4 is not -2, it is +2. y 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. Using bsolute Value Slide 48 (nswer) / 87 Take -2 for example. MP.6: ttend to precision (-2) 2 = +4 Emphasize the use of parentheses when ut, 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 y 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. Math Practice 4 = 2 This accounts for +2 2 and (-2) 2.

27 Using bsolute Value Slide 49 / 87 Easy enough. ut 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". Simplifying Roots of Variables Slide 50 / 87 The technical definition of "the square root of x squared" is "the absolute value of x". x x x x - - = = x 2 = x 2 x 2 x x is positive x is negative Simplifying Roots of Variables Using bsolute Values Slide 51 / 87 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

28 ut, Why? Slide 52 / 87 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 Roots of Variable Practice Slide 53 / 87 More Examples Use expanded form to explain why absolute value must be used in these answers. Roots of Variable Practice Slide 53 (nswer) / 87 Math Practice More Examples Use expanded form to explain why absolute MP.3: onstruct value viable must arguments be used in and these answers. critique the reasoning of others. MP.7: Look for and make use of structure. sk: 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)

29 Simplifying Roots of Variables Slide 54 / 87 ivide 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: bsolute value signs are not needed because the radicand had an odd power to start. Examples: Roots of Variables Examples Slide 55 / 87 ombining it all: 50x 4 y 12 z (x 2 ) 2 (y 6 ) 2 z zz 5 x 2 y 6 z 2z Roots of Variables Practice Slide 56 / 87 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.

30 Slide 57 / 87 Slide 57 (nswer) / 87 Slide 58 / 87

31 Slide 58 (nswer) / Simplify Slide 59 / Simplify Slide 59 (nswer) / 87 nswer

32 27 Simplify Slide 60 / Simplify Slide 60 (nswer) / 87 nswer Slide 61 / 87 Solving Equations with Perfect Square and ube Roots Return to Table of ontents

33 Slide 62 / 87 Squares and ubes Practice Use the numbers shown to make the equations true. Each number can be used only once. (Problem from ) Slide 63 / a. = b. 3 = Squares and ubes Practice Use the numbers shown to make the equations true. Each number can be used only once. (Problem from ) Slide 63 (nswer) / nswer a. = b. 3 =

34 Squares and ubes Practice omplete the Venn-iagram to classify the numbers as perfect squares and perfect cubes (Problem from ) Slide 64 / 87 Perfect Squares Perfect ubes Squares and ubes Practice omplete the Venn-iagram to classify the numbers as perfect squares and perfect cubes (Problem from ) Slide 64 (nswer) / 87 nswer Perfect Squares Perfect ubes Solving Equations Slide 65 / 87 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.

35 Solving Equations Example Slide 66 / 87 Example: Solve. = ivide each side by the coefficient. Then take the square root of each side. Example: Solve. Solving Equations Example Slide 67 / 87 Multiply each side by nine, then take the cube root of each side. Notice! Slide 68 / 87 The answer is only a positive 3, not Why is the answer only positive and not both positive and negative?

36 ube Roots Slide 69 / 87 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 N take the cube root of a negative number! ube Roots Examples Slide 70 / 87 Squares and ubes Practice Slide 71 / 87 Try These: Solve. ± 10 ± 8 ± 9 ± 7

37 Try These: Squares and ubes Practice Slide 72 / 87 Solve Solve. Slide 73 / Solve. Slide 73 (nswer) / 87 nswer ±12

38 29 Solve. Slide 74 / Solve. Slide 74 (nswer) / 87 nswer ±12 30 Solve. Slide 75 / 87

39 30 Solve. Slide 75 (nswer) / 87 nswer 2 31 Solve. Slide 76 / Solve. Slide 76 (nswer) / 87 nswer 4

40 32 Solve 15 + x 2 = 40 Slide 77 / 87 erived from 32 Solve 15 + x 2 = 40 Slide 77 (nswer) / 87 nswer ±5 erived from 33 Solve 2 + x 3 = 10 Slide 78 / 87 erived from

41 33 Solve 2 + x 3 = 10 Slide 78 (nswer) / 87 nswer 2 erived from 34 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. Slide 79 / 87 erived from 34 cube has a volume of 343 cm 3. a) Write an equation that could be used to determine the length, L, of one side. Slide 79 (nswer) / 87 b) Solve the equation. nswer a) L 3 = 343 b) L = 7 cm erived from

42 35 Estimate the area of the rectangle to the nearest tenth. Slide 80 / Estimate the area of the rectangle to the nearest tenth. Slide 80 (nswer) / 87 nswer u 2 36 If the area of a square is square inches, what is the length, in inches, of one side of the square? Slide 81 / 87

43 36 If the area of a square is square inches, what is the length, in inches, of one side of the square? Slide 81 (nswer) / 87 nswer 37 Which equation has both 4 and -4 as possible values of y? Slide 82 / 87 From PR EOY sample test non-calculator #9 37 Which equation has both 4 and -4 as possible values of y? Slide 82 (nswer) / 87 nswer From PR EOY sample test non-calculator #9

44 Slide 83 / 87 Glossary & Standards Return to Table of ontents ube Slide 84 / 87 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 ack to Instruction ube Root value that, when used in a multiplication three times, gives that number. Slide 85 / 87 Symbol: 3 "cube root" 3 64 = 4 (4)(4)(4) = 64 4x4x4 = = 6 (6)(6)(6) = 216 6x6x6 = 216 ack to Instruction

45 Power power is another name for an exponent. It is a small, raised number that shows how many times to multiply the base by itself. Slide 86 / 87 Power 3 2 ase "3 to the second power" 3 2 = 3x = x x x x 3 ack to Instruction Standards for Mathematical Practice Slide 87 / 87 MP1 Making sense of problems & persevere in solving them. MP2 Reason abstractly & quantitatively. MP3 onstruct viable arguments and critique the reasoning of others. MP4 Model with mathematics. MP5 Use appropriate tools strategically. MP6 ttend to precision. MP7 Look for & make use of structure. MP8 Look for & express regularity in repeated reasoning. lick on each standard to bring you to an example of how to meet this standard within the unit.