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 Contents Click on topic to go to that section. Radical Expressions Containing Variables Simplifying Non-Perfect Square Radicands Simplifying Roots of Variables Radical Expressions Solving Equations with Perfect Square & Cube Roots Glossary & Standards Containing Variables Return to Table of Contents Slide 5 / 87 Slide 6 / 87 Square Roots of Variables Square Roots of Variables To take the square root of a variable rewrite its exponent as the If the square root of a variable raised to an even power has a square of a power. 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... (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?
Slide 7 / 87 Slide 8 / 87 Square Root Practice Square Root Practice Try These. Examples = |x| 5 = |x| 13 Slide 9 / 87 Slide 10 / 87 Square Root Practice How many of these expressions will need an absolute value sign when simplified? yes yes no no yes yes Slide 11 / 87 Slide 12 / 87
Slide 13 / 87 Slide 14 / 87 5 C A B D no real solution Slide 15 / 87 Slide 16 / 87 Simplifying Perfect Squares (Review) A number is a perfect square if you can Unit 1 take that quantity of 1x1 unit squares Square and form them into a square. 1 Simplifying Non-Perfect 4 is a perfect square, because you can take 4 Square Radicands unit squares and form them into a 2x2 square. 2 (Notice that the square root of 4 is the length of one 2 of its sides, since that side times itself equals 4.) 4 = 2 Return to Table of Contents Slide 17 / 87 Slide 18 / 87 Non-Perfect Squares Non-Perfect Squares What About Numbers that are not Perfect Squares? What happens when the radicand is not a perfect square? How can we simplify ? 8 8 Rewrite the radicand as a product of its largest perfect square factor. click 8 = 2 2 2 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. Simplify the square root of the perfect square. click So, we know that we will not have a whole number, which we can multiply by itself, to equal 8. When simplified form still contains a radical, it is said to be irrational.
Slide 19 / 87 Slide 20 / 87 Simplifying Non-Perfect Squares Non-Perfect Squares Identifying the largest perfect square factor when simplifying radicals will result in the least amount of work. What happens when the radicand is not a perfect square? 1. Rewrite the radicand as a product of its largest perfect Ex: square factor. Not simplified! Keep going! 2. Simplify the square root of the perfect square. Finding the largest perfect square factor results in less work: click click click When simplified form still contains a radical, Note that the answers are the same for both solution processes it is said to be irrational. Slide 21 / 87 Slide 22 / 87 Simplifying Non-Perfect Squares Simplifying Non-Perfect Squares Another method for simplifying non-perfect squares is to use 48 prime factorization and a factor tree. For example, 48 can be broken down as follows: 2 24 2(2) 3 = 4 3 2 12 48 2 6 2 24 2 3 2 12 After you factor the number into all of its primes, you can circle 2 6 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 2 3 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. Slide 23 / 87 Slide 24 / 87 Non-Perfect Squares Practice 6 Simplify Try These. A B C D already in simplified form
Slide 25 / 87 Slide 26 / 87 7 Simplify 8 Simplify A A B B C C D already in simplified D already in simplified form form Slide 27 / 87 Slide 28 / 87 9 Simplify 10 Simplify A A B B C C D already in simplified D already in simplified form form Slide 29 / 87 Slide 30 / 87 11 Simplify 12 Which of the following does not have an irrational simplified form? A B A C B D already in simplified form C D
Slide 31 / 87 Slide 32 / 87 13 The diagonal of a square can be expressed by the 14 The distance, d , in miles that a person can see to the formula d= 2 a 2 , where a is the side length of the square. horizon is calculated with the following formula. Select the correct options to show the length of the diagonal of the square shown. h = the person's height 3h d = Your answer should be a radicand above sea level in feet. 2 in simplest form. How far to the horizon would you be able to see from this vantage point? 100 ft above sea level Your answer should be a radicand in 9 simplest form. d = ___ ___ d = ___ ___ A 3 D 1 D 5 A 3 B 4 E 2 E 6 B 4 C 5 F 10 C 9 F 3 Slide 33 / 87 Slide 34 / 87 Simplest Radical Form Simplest Radical Form Note - If a radical begins with a coefficient before the radicand Likewise - If a radical begins with a coefficient before the is simplified, any perfect square that is simplified will be radicand is simplified, any pair of primes that are circled will be multiplied by the existing coefficient. (multiply the outside) multiplied by the existing coefficient. (multiply the outside) 2 2 18 7 12 2 6 2 9 3 3 2 3 2(3) 2 7(2) 3 6 2 14 3 Slide 35 / 87 Slide 36 / 87 15 Simplify A B C D
Slide 37 / 87 Slide 38 / 87 16 Simplify 17 Simplify A A B B C C D D Slide 39 / 87 Slide 40 / 87 18 Simplify 19 Simplify A A B B C C D D Slide 41 / 87 Slide 42 / 87 20 When is written in simplest radical form, the result is . What is the value of k? A 20 Teachers: B 10 Use the questions found in the pull tab for the C 7 next 2 slides. 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.
Slide 43 / 87 Slide 44 / 87 21 When is expressed in simplest 22 Which is greater or 6? 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 www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. Derived from Slide 45 / 87 Slide 46 / 87 23 Which is greater or 10? Simplifying Roots of Variables Return to Table of Derived from Contents Slide 47 / 87 Slide 48 / 87 Using Absolute Value Using Absolute Value Take -2 for example. When we simplify radicals, we are told to assume (-2) 2 = +4 all variables are positive. But, why? But, 4 is not -2, it is +2. Because, the square root of the square of a negative number is not the original number. 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 .
Slide 49 / 87 Slide 50 / 87 Using Absolute Value Simplifying Roots of Variables Easy enough. The technical definition of But what about when the radicand is a variable, "the square root of x squared" is "the absolute value of x". and we don't know the sign of the unknown value? x 2 = x x 2 Is x positive or negative? x x x 2 x is positive = x x x 2 We can't know, so we - - = x is negative "assume all variables are positive". Slide 51 / 87 Slide 52 / 87 But, Why? Simplifying Roots of Variables Using Absolute Values x 6 = x 3 When working with square roots, an absolute value sign is needed if: = x x x x x x x x x The power of the given variable is even. x 6 · Whether x is positive or negative, However, if x is negative, when it when it is multiplied by itself an is multiplied by itself an odd and even number of times, it will turn number of times, it will turn out to The answer contains a variable raised to an x 3 out to be a positive number. be a negative number. · odd power outside the radical. So, x is positive. So, x could be negative. x 6 = x 3 x 6 = x 3 So, in order for , we must use an absolute value sign to indicate that x is positive. x 6 = x 3 Slide 53 / 87 Slide 54 / 87 Roots of Variable Practice Simplifying Roots of Variables More Examples Divide the exponent by 2. The number of times that 2 goes into the Use expanded form to explain why absolute exponent becomes the power on the outside of the radical and the value must be used in these answers. remainder is the power of the radicand. x x x x x x x x x 7 = x 3 = Note: Absolute value signs are not needed because the radicand had an odd power to start.
Slide 55 / 87 Slide 56 / 87 Roots of Variables Examples Roots of Variables Practice Examples: 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. Combining it all: 50 x 4 y 12 z The m's starting power was odd, so it does not require absolute 3 value signs. z z z 25 2( x 2 ) 2 ( y 6 ) 2 5 x 2 y 6 z 2z Slide 57 / 87 Slide 58 / 87 Slide 59 / 87 Slide 60 / 87 26 Simplify 27 Simplify A A B B C C D D
Recommend
More recommend
