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8th Grade Equations with Roots and Radicals 2015 12 17 www.njctl.org 2
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 3
Radical Expressions Containing Variables Return to Table of Contents 4
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 Answer & Math Practice = (a 8 ) 2 = a 8 Can you find a shortcut to solve this type of problem? How would your shortcut make the problem easier? 5
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... 6
Square Root Practice Examples 7
Square Root Practice Try These. = x 5 = x 13 8
Square Root Practice How many of these expressions will need an absolute value sign when simplified? yes yes no no yes yes 9
1 Simplify A B C D Answer 10
2 Simplify A B C D Answer 11
3 Simplify A B C D Answer 12
4 Simplify A B C D Answer 13
5 A C B D no real solution Answer 14
Simplifying Non Perfect Square Radicands Return to Table of Contents 15
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. 2 (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 16
Non Perfect Squares What About Numbers that are not Perfect Squares? How can we simplify 8? Math Practice 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. 17
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 2 2 2 = 2 2 When simplified form still contains a radical, it is said to be irrational. 18
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. 19
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 20
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: 48 2 24 2 12 2 6 2 3 21
Simplifying Non Perfect Squares 48 2 24 2 12 2 6 2 3 2(2) 3 = 4 3 Teacher Notes 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. 22
Try These. Non Perfect Squares Practice Prime Factoring Answer 23
6 Simplify A B C D already in simplified form Answer 24
7 Simplify A B C D already in simplified form Answer 25
8 Simplify A B C D already in simplified form Answer 26
9 Simplify A B C D already in simplified form Answer 27
10 Simplify A B C D already in simplified form Answer 28
11 Simplify A B C D already in simplified form Answer 29
12 Which of the following does not have an irrational simplified form? A B C Answer D 30
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. d = 9 Answer A 3 D 1 B 4 E 2 C 9 F 3 31
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. 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 Answer d = A 3 B 4 C 5 D 5 E 6 F 10 32
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 33
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 18 2 9 3 3 2(3) 2 6 2 7 12 2 6 2 3 7(2) 3 14 3 34
Simples Radical Form Practice Express in simplest radical form. Math Practice 35
15 Simplify A B C D Answer 36
16 Simplify A B C D Answer 37
17 Simplify A B C D Answer 38
18 Simplify A B C D Answer 39
19 Simplify A B C D Answer 40
Teachers: Use the questions found in the pull tab for the next 2 slides. Math Practice 41
20 When is written in simplest radical form, the result is. What is the value of k? A 20 B 10 C 7 Answer D 4 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011. 42
21 When is expressed in simplest form, what is the value of a? A 6 B 2 C 3 D 8 Answer From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011. 43
22 Which is greater or 6? Answer Derived from 44
23 Which is greater or 10? Answer Derived from 45
Simplifying Roots of Variables Return to Table of Contents 46
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. 47
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. Math Practice 4 = 2 This accounts for +2 2 and ( 2) 2. 48
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". 49
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 50
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 51
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 52
Roots of Variable Practice More Examples Use expanded form to explain why absolute value must be used in these answers. Math Practice 53
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. 54
Examples: Roots of Variables Examples Combining it all: 50x 4 y 12 z 3 25 2(x 2 ) 2 (y 6 ) 2 z zz 5 x 2 y 6 z 2z 55
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. 56
24 Simplify A B C D Answer Hint Remember so 57
25 Simplify A B C D Answer 58
26 Simplify A B C D Answer 59
27 Simplify A B C Answer D 60
Solving Equations with Perfect Square and Cube Roots Return to Table of Contents 61
Squares and Cubes The product of two equal factors is the "square" of the number. The product of three equal factors is the "cube" of the number. 62
Squares and Cubes Practice Use the numbers shown to make the equations true. Each number can be used only once. (Problem from ) 4 8 10 64 1000 100 a. = Answer b. 3 = 63
Squares and Cubes Practice Complete the Venn Diagram to classify the numbers as perfect squares and perfect cubes. 1 64 96 125 200 256 333 361 (Problem from ) Answer Perfect Squares Perfect Cubes 64
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 = # 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. 65
Solving Equations Example Example: Solve. = Divide each side by the coefficient. Then take the square root of each side. 66
Example: Solving Equations Example Solve. Multiply each side by nine, then take the cube root of each side. 67
Notice! The answer is only a positive 3, not 3. + Why is the answer only positive and not both positive and negative? 68
Cube 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... 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 AND take the cube root of a negative number! 69
Cube Roots Examples 70
Try These: Solve. Squares and Cubes Practice ± 10 ±8 ±9 ±7 71
Try These: Solve. Squares and Cubes Practice 2 1 4 5 72
28 Solve. Answer 73
29 Solve. Answer 74
30 Solve. Answer 75
31 Solve. Answer 76
32 Solve 15 + x 2 = 40 Answer Derived from 77
33 Solve 2 + x 3 = 10 Answer Derived from 78
34 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 Derived from 79
35 Estimate the area of the rectangle to the nearest tenth. Answer 80
36 If the area of a square is square inches, what is the length, in inches, of one side of the square? A B C Answer D 81
37 Which equation has both 4 and 4 as possible values of y? A B C Answer D From PARCC EOY sample test non calculator #9 82
Glossary & Standards Return to Table of Contents 83
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 84
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 = 64 3 216 = 6 (6)(6)(6) = 216 6x6x6 = 216 Back to Instruction 85
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 = 3 x 3 3 3 = 3x 3x 3 3 2 x 2 3 3 3 x 3 3 Back to Instruction 86
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. 87