Slide 1 / 87 Slide 2 / 87 8th Grade Equations with Roots and Radicals 2015-12-17 www.njctl.org Slide 3 / 87 Slide 4 / 87 Table of ontents 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. Radical Expressions ontaining Variables Return to Table of ontents Slide 5 / 87 Square Roots of Variables To take the square root of a variable rewrite its exponent as the square of a power. Slide 5 (nswer) / 87 Square Roots of Variables To take the square root of a variable rewrite its exponent as the square of a power. nswer: ivide the exponent inside of the square root by 2. nswer & Math Practice = = (x 12 ) 2 = x 12 (a 8 ) 2 = a 8 = = (x 12 ) 2 = x 12 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? an you find a shortcut to solve this type of problem? How would your shortcut make the problem easier?
Slide 6 / 87 Square Roots of Variables Slide 7 / 87 Square Root Practice 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. Examples y efinition... Try These. = x 5 Slide 8 / 87 Square Root Practice Slide 9 / 87 Square Root Practice How many of these expressions will need an absolute value sign when simplified? yes yes = x 13 no no yes yes Slide 10 / 87 Slide 10 (nswer) / 87
Slide 11 / 87 Slide 11 (nswer) / 87 Slide 12 / 87 Slide 12 (nswer) / 87 Slide 13 / 87 Slide 13 (nswer) / 87
Slide 14 / 87 Slide 14 (nswer) / 87 5 5 no real solution nswer no real solution Slide 15 / 87 Simplifying Non-Perfect Square Radicands Return to Table of ontents Slide 16 / 87 Simplifying Perfect Squares (Review) number is a perfect square if you can take that quantity of 1x1 unit squares and them into a square. 4 is a perfect square, because you can take 4 unit squares and them into a 2x2 square. 1 1 Unit 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 Slide 17 / 87 Non-Perfect Squares 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 them into a square. So, we know that we will not have a whole number, which we can multiply by itself, to equal 8. Math Practice Slide 17 (nswer) / 87 Non-Perfect Squares 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 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.
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 = 2 2 2 Simplify the square root of the perfect square. click 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. When simplified still contains a radical, it is said to be irrational. click click click When simplified still contains a radical, it is said to be irrational. Slide 20 / 87 Identifying the largest perfect square factor when simplifying radicals will result in the least amount of work. Ex: Simplifying Non-Perfect Squares Not simplified! Keep going! Finding the largest perfect square factor results in less work: Slide 21 / 87 Simplifying Non-Perfect Squares 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: 48 2 24 2 12 Note that the answers are the same for both solution processes 2 6 2 3 Slide 22 / 87 Simplifying Non-Perfect Squares Slide 22 (nswer) / 87 Simplifying Non-Perfect Squares 48 2 24 2 12 2 6 2 3 2(2) 3 = 4 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. 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.
Try These. Slide 23 / 87 Non-Perfect Squares Practice Try These. Prime Factoring nswer Slide 23 (nswer) / 87 Non-Perfect Squares Practice 72 2 36 2 18 2 9 3 3 2(3) 2 6 2 360 2 180 2 90 2(3) 2(5) 2 45 6 10 3 15 3 5 18 2 9 3 3 3 2 24 2 12 2 6 2 3 2 6 Slide 24 / 87 Slide 24 (nswer) / 87 6 Simplify 6 Simplify already in simplified nswer already in simplified Slide 25 / 87 Slide 25 (nswer) / 87 7 Simplify 7 Simplify already in simplified nswer already in simplified
Slide 26 / 87 Slide 26 (nswer) / 87 8 Simplify 8 Simplify already in simplified nswer already in simplified Slide 27 / 87 Slide 27 (nswer) / 87 9 Simplify 9 Simplify already in simplified nswer already in simplified Slide 28 / 87 Slide 28 (nswer) / 87 10 Simplify already in simplified 10 Simplify nswer already in simplified
Slide 29 / 87 Slide 29 (nswer) / 87 11 Simplify 11 Simplify already in simplified nswer already in simplified Slide 30 / 87 12 Which of the following does not have an irrational simplified? Slide 30 (nswer) / 87 12 Which of the following does not have an irrational simplified? nswer Slide 31 / 87 13 The diagonal of a square can be expressed by the ula 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. Slide 31 (nswer) / 87 13 The diagonal of a square can be expressed by the ula 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. d = 9 d = nswer, E 9 3 1 3 1 4 E 2 4 E 2 9 F 3 9 F 3
Slide 32 / 87 Slide 32 (nswer) / 87 14 The distance, d, in miles that a person can see to the horizon is calculated with the following ula. 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. h = the person's height above sea level in feet. 100 ft above sea level 14 The distance, d, in miles that a person can see to the horizon is calculated with the following ula. 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. nswer h = the person's height above 300 sea is level divisible in feet. by 2. So, 100 ft above sea level 300 2 = 150 d = 3 4 5 E 6 d = 3 4 5 E 6, E 5 F 10 5 F 10 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) 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) 2 2 18 2 9 3 3 2(3) 2 6 2 7 12 2 6 2 3 7(2) 3 14 3 Slide 35 / 87 Slide 35 (nswer) / 87
Slide 36 / 87 Slide 36 (nswer) / 87 15 Simplify 15 Simplify nswer Slide 37 / 87 Slide 37 (nswer) / 87 16 Simplify 16 Simplify nswer Slide 38 / 87 Slide 38 (nswer) / 87 17 Simplify 17 Simplify nswer
Slide 39 / 87 Slide 39 (nswer) / 87 18 Simplify 18 Simplify nswer Slide 40 / 87 Slide 40 (nswer) / 87 19 Simplify 19 Simplify nswer Slide 41 / 87 Slide 41 (nswer) / 87 Teachers: Use the questions found in the pull tab for the next 2 slides. 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
Slide 42 / 87 20 When is written in simplest radical, the result is. What is the value of k? Slide 42 (nswer) / 87 20 When is written in simplest radical, the result is. What is the value of k? 20 20 10 7 10 7 nswer 4 4 From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011. From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011. Slide 43 / 87 Slide 43 (nswer) / 87 21 When is expressed in simplest, what is the value of a? 21 When is expressed in simplest, what is the value of a? 6 6 2 3 8 2 3 8 nswer From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011. From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011. Slide 44 / 87 Slide 44 (nswer) / 87 22 Which is greater or 6? 22 Which is greater or 6? nswer 6 erived from erived from
23 Which is greater or 10? Slide 45 / 87 23 Which is greater or 10? Slide 45 (nswer) / 87 nswer 10 erived from erived from Slide 46 / 87 Slide 47 / 87 Using bsolute Value Simplifying Roots of Variables 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. Return to Table of ontents Take -2 for example. (-2) 2 = +4 ut, 4 is not -2, it is +2. Slide 48 / 87 Using bsolute Value 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. Slide 48 (nswer) / 87 Using bsolute Value 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 = -(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. 4 = 2 This accounts for +2 2 and (-2) 2.
Slide 49 / 87 Using bsolute Value Easy enough. ut what about when the radicand is a variable, and we don't know the sign of the unknown value? x 2 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 Is x positive or negative? We can't know, so we "assume all variables are positive". x x x x - - = = x 2 x 2 x is positive x is negative Slide 51 / 87 Simplifying Roots of Variables Using bsolute Values When working with square roots, an absolute value sign is needed if: Slide 52 / 87 ut, Why? x 6 = x 3 x x x x x x = x x x 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 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. x 6 = x 3 Slide 53 / 87 Roots of Variable Practice More Examples Use expanded to explain why absolute value must be used in these answers. Math Practice 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 Slide 53 (nswer) / 87 Roots of Variable Practice More Examples Use expanded 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)
Slide 54 / 87 Simplifying Roots of Variables 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. Examples: Slide 55 / 87 Roots of Variables Examples x 7 = x x x x x x x = x 3 x ombining it all: Note: bsolute value signs are not needed because the radicand had an odd power to start. 50x 4 y 12 z 3 25 2(x 2 ) 2 (y 6 ) 2 z zz 5 x 2 y 6 z 2z Slide 56 / 87 Slide 57 / 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. Slide 57 (nswer) / 87 Slide 58 / 87
Slide 58 (nswer) / 87 Slide 59 / 87 26 Simplify Slide 59 (nswer) / 87 Slide 60 / 87 26 Simplify 27 Simplify nswer Slide 60 (nswer) / 87 Slide 61 / 87 27 Simplify nswer Solving Equations with Perfect Square and ube Roots Return to Table of ontents
Slide 62 / 87 Slide 63 / 87 Squares and ubes Practice Use the numbers shown to make the equations true. Each number can be used only once. (Problem from ) 4 8 10 64 100 1000 a. = b. 3 = Slide 63 (nswer) / 87 Squares and ubes Practice Use the numbers shown to make the equations true. Each number can be used only once. (Problem from ) 4 8 10 64 100 1000 Slide 64 / 87 Squares and ubes Practice omplete the Venn-iagram to classify the numbers as perfect squares and perfect cubes. 1 64 96 125 200 256 333 361 (Problem from ) nswer a. = b. 3 = Slide 64 (nswer) / 87 Squares and ubes Practice omplete the Venn-iagram to classify the numbers as perfect squares and perfect cubes. 1 64 96 125 200 256 333 361 (Problem from ) Perfect Squares Perfect ubes 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: Solving Equations x 2 = # nswer you must find the square root of both sides in order to find the value of x. When your equation simplifies to: x 3 = # Perfect Squares Perfect ubes you must find the cube root of both sides in order to find the value of x.
Slide 66 / 87 Slide 67 / 87 Solving Equations Example Solving Equations Example Example: Example: Solve. = ivide each side by the coefficient. Then take the square root of each side. Solve. Multiply each side by nine, then take the cube root of each side. Slide 68 / 87 Slide 69 / 87 Notice! ube Roots The cube root of 27 is 3, and not -3, because when 3 is cubed you get 27. 3 x 3 x 3 = 27 If you were to cube -3, you would get -27... The answer is only a positive 3, not 3. + - Why is the answer only positive and not both positive and negative? -3 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! Slide 70 / 87 ube Roots Examples Slide 71 / 87 Squares and ubes Practice Try These: Solve. ± 10 ± 8 ± 9 ± 7
Slide 72 / 87 Slide 73 / 87 Try These: Squares and ubes Practice 28 Solve. Solve. 2 1 4 5 Slide 73 (nswer) / 87 Slide 74 / 87 28 Solve. 29 Solve. nswer ±12 Slide 74 (nswer) / 87 Slide 75 / 87 29 Solve. 30 Solve. nswer ±12
Slide 75 (nswer) / 87 Slide 76 / 87 30 Solve. 31 Solve. nswer 2 Slide 76 (nswer) / 87 Slide 77 / 87 31 Solve. 32 Solve 15 + x 2 = 40 nswer 4 erived from Slide 77 (nswer) / 87 Slide 78 / 87 32 Solve 15 + x 2 = 40 33 Solve 2 + x 3 = 10 nswer ±5 erived from erived from
33 Solve 2 + x 3 = 10 Slide 78 (nswer) / 87 34 Slide 79 / 87 cube has a volume of 343 cm 3. a) Write an equation that could be used to determine the length, L, of one side. nswer 2 b) Solve the equation. erived from erived from Slide 79 (nswer) / 87 Slide 80 / 87 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. 35 Estimate the area of the rectangle to the nearest tenth. b) Solve the equation. nswer a) L 3 = 343 b) L = 7 cm erived from Slide 80 (nswer) / 87 35 Estimate the area of the rectangle to the nearest tenth. Slide 81 / 87 36 If the area of a square is square inches, what is the length, in inches, of one side of the square? nswer 220.5 u 2
Slide 81 (nswer) / 87 36 If the area of a square is square inches, what is the length, in inches, of one side of the square? nswer Slide 82 / 87 37 Which equation has both 4 and -4 as possible values of y? From PR EOY sample test non-calculator #9 Slide 82 (nswer) / 87 Slide 83 / 87 37 Which equation has both 4 and -4 as possible values of y? nswer Glossary & Standards From PR EOY sample test non-calculator #9 Return to Table of ontents To multiply a number by itself and then again by itself. Slide 84 / 87 ube The product of three equal factors. Slide 85 / 87 ube Root value that, when used in a multiplication three times, gives that number. 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 Symbol: 3 "cube root" 3 64 = 4 (4)(4)(4) = 64 4x4x4 = 64 3 216 = 6 (6)(6)(6) = 216 6x6x6 = 216 ack to Instruction ack to Instruction
Power 3 2 ase Slide 86 / 87 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. "3 to the second power" 3 2 = 3x 3 3 3 = x x 3 3 3 3 2 3 x 2 3 3 3 x 3 ack to Instruction Slide 87 / 87 Standards for Mathematical Practice 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.