Unit 4 notes

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Unit 4 notes

  1. 1. Unit 4 Notebook 2012.notebook December 04, 2012 Simplifying Radicals Irrational Expressions Nov 5­9:17 AM Nov 5­9:17 AM Do you know your perfect squares??? List the first 15 perfect squares... Did you know... 1 81 4 100 9 121 16 144 25 169 36 196 49 225 64 Nov 5­9:17 AM Nov 5­9:17 AM 1 1 4 4 Can you simplify the following rational expressions?   9 16 What if the expressions were IRRATIONAL?  Do you  9 16 Explain why each is rational! 25 36 know how to simplify these? 25 36 49 49 64 64 81 81 100 100 121 121 144 144 169 169 196 196 225 225 ... ... Nov 5­9:17 AM Nov 5­9:17 AM 1
  2. 2. Unit 4 Notebook 2012.notebook December 04, 2012 Steps to Simplifying Radicals: How would you simplify a radical that has a variable inside of it??? 1. Rewrite the irrational expression as the Can you simplify the following: product of 2 radicals. One of those radicals MUST have a perfect square inside it. 2. Break down the radical with the perfect square inside it. 3. Write your final answer. Nov 5­9:17 AM Nov 5­9:17 AM Can you simplify the following irrational expressions? Can you simplify the following irrational expressions? 1 1 4 4 9 9 16 16 25 25 36 36 49 49 64 64 81 81 100 100 121 121 144 144 169 169 196 196 225 225 ... ... Nov 5­9:17 AM Nov 5­9:17 AM #3 #7 Homework: p. 1‐2 #1‐ 21 odds #11 #15 Nov 5­9:17 AM Nov 5­9:17 AM 2
  3. 3. Unit 4 Notebook 2012.notebook December 04, 2012 Answers to p. 1‐ 2 #1‐ 21 odds Rationalizing Radicals 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21. Nov 5­9:17 AM Nov 5­9:19 AM 1 1 So we know how to simplify ... 4 9 Rationalizing a denominator: 4 16 Removing all radical symbols from the 9 16 25 36 denominator of a fraction. 25 49 36 64 49 81 100 Does anyone know the best way to get rid of the 64 81 121 radical symbol in our example? 100 Do you know WHY we have to simplify 144 121 169 144 196 169 225 196 ... 225 ... Nov 5­9:19 AM Nov 5­9:19 AM 1 Rationalizing each expression below: 4 9 16 25 Steps to RATIONALIZING a fraction: 36 49 1. Simplify the expression if possible by dividing. 64 2. Multiply the top and bottom by the radical in 81 100 the denominator. 121 144 3. Simplify the remaining fraction. 169 196 225 ... Nov 5­9:19 AM Nov 5­9:19 AM 3
  4. 4. Unit 4 Notebook 2012.notebook December 04, 2012 1 Rationalizing each expression below: 4 9 16 25 36 49 64 81 100 121 Homework: 144 169 196 225 ... p. 3 #1- 8 Nov 5­9:19 AM Nov 5­9:19 AM #2 #4 Answers to p. 3 #1- 8 1. 2. 3. 4. #6 #8 5. 6. 7. 8. Nov 5­9:19 AM Nov 5­9:19 AM Do you recall how to simplify the following: Adding & Subtracting Radicals 1. 2x + 3x 2. -5xy - 8xy 3. 7x2y - 17x2y + 9xy2 Nov 5­9:20 AM Nov 5­9:20 AM 4
  5. 5. Unit 4 Notebook 2012.notebook December 04, 2012 Then you should be able to simplify the When you add or subtract radicals, you following: must have the same # inside the radical 1. symbol (radicand). 1 4 Its just like combining like terms!!! 9 16 25 2. 36 49 Ex: 64 81 100 121 144 3. 169 196 225 ... Nov 5­9:20 AM Nov 5­9:20 AM 1 Use a bit of logic to do these problems. Simplify Simplify: 4 9 the radical that is easier to YOU first! Then you 1. 2. 16 25 know what the "last name" has to be for the other 36 49 radical. 64 81 100 121 144 Ex: 169 3. 4. 196 225 Which radical is easier to break ... down? What is the "last name" going to be? 5. 6. Nov 5­9:20 AM Nov 5­9:20 AM #9 #15 Homework: p. 4‐ 5 # 7‐ 27 odds #17 #21 Nov 5­9:20 AM Nov 5­9:20 AM 5
  6. 6. Unit 4 Notebook 2012.notebook December 04, 2012 Answers to p. 4 ‐5, #7‐ 27 odds QUIZ #6 TODAY 1 4 9 If you can simplify the following problems, then you should be ok ;) 16 25 36 7. 9. 11. 1. 2. 3. 4. 49 64 81 100 121 13. 15. 17. 144 169 196 225 ... 19. 21. 23. 25. 27. Nov 5­9:20 AM Nov 5­9:20 AM QUIZ #6­ Do NOT do #2, 4 & 7 Journal Entry- Friday, Nov. 30 1 4 9 16 25 CR #6 due tomorrow Would you like to write a short paper on a 36 49 mathematician and create a poster based on their 64 "findings" to count as a test? 81 100 BONUS (3 points) 121 144 169 Simplify: Would you like to work in groups (3 people 196 225 max) or by yourself? ... HAPPY FRIDAY! Nov 5­9:20 AM Nov 5­9:20 AM Steps to Multiplying Radicals: Multiplying 1. Multiply OUTSIDES with OUTSIDES Radicals 2. Multiply INSIDES with INSIDES. 3. Simplify if possible. Wh at # s a re o n th Example: Wh at 3 s a re o eO UTS IDE n th ? e IN SID E? Nov 5­9:20 AM Nov 5­9:20 AM 6
  7. 7. Unit 4 Notebook 2012.notebook December 04, 2012 Try these. Multiply and simplify: 1 Uh oh‐ these have variables! Multiply and 1 4 9 simplify: 4 9 16 16 25 1. 2. 36 25 36 49 64 1. 2. 49 64 81 81 100 100 121 121 144 144 169 169 196 196 3. 4. 225 ... 225 3. 4. ... Nov 5­9:20 AM Nov 5­9:20 AM Using the Distributive Property: 1 4 (Remember, OUTSIDES with OUTSIDES, INSIDES with INSIDES) 9 16 1. 2. 25 36 49 64 81 Homework: p. 6 # 1‐ 14 100 121 144 169 196 225 3. 4. ... Nov 5­9:20 AM Nov 5­9:20 AM p. 6 #1­ 14 #3 #6 Answers to p. 6 # 1‐ 14 1. 2. 3. 4. 5. 6. #9 #12  7. 8. 9. 10. 11. 12. 13. 14. Nov 5­9:20 AM Nov 5­9:20 AM 7
  8. 8. Unit 4 Notebook 2012.notebook December 04, 2012 FOIL-ing Radicals Do you recall how to multiply binomials? You guys call this the FOIL-ing Method. Multiply and simplify: 1. (x + 2) (x - 9) = 2. (3 - x) (3 + x) = 3. (x2 - 1) (x2 - 7) = Nov 5­9:21 AM Nov 5­9:21 AM Can you apply the FOIL-ing Method to Multiply and simplify: RADICALS??? 1. 2. Remember OUTSIDES with OUTSIDES and INSIDES with INSIDES! 1 1 4 4 9 9 16 Multiply and simplify: 16 25 25 36 1. = 36 49 49 64 64 81 81 100 100 121 3. 121 144 144 169 169 196 196 225 225 ... ... 2. = Nov 5­9:21 AM Nov 5­9:21 AM Answers to p. 7 #21­ 28 21. 22. Homework: 23. 24. p. 7 #21­ 28 25. 26. 27. 28. Nov 5­9:21 AM Nov 5­9:21 AM 8
  9. 9. Unit 4 Notebook 2012.notebook December 04, 2012 Activity? Pythagorean Theorem Nov 5­9:21 AM Nov 5­9:21 AM y 10 9 8 C Mrs. E cant find her car in the parking lot 7 6 after a long trip at Target. Mrs. E is Mrs. E cant find her car in the parking lot 5 4 standing at T(-4, 2). After much ado, she after a long trip at Target. Mrs. E is T 3 2 sees her car is parked at C (4, 8). Can you standing at T(-4, 2). After much ado, she 1 x determine how far away from her car she sees her car is parked at C (4, 8). Can you ­10 ­8 ­6 ­4 ­2 0 2 4 6 8 10 is??? determine how far away from her car she ­2 ­3 is??? ­4 ­5 ­6 ­7 ­8 ­9 ­10 Nov 5­9:21 AM Nov 5­9:21 AM Pythagoras (569-500 B.C.E.) was born on the island of Samos in Greece, and did much traveling through Egypt, learning, among other things, mathematics. DID YOU KNOW??? Not much more is known of his early years. Pythagoras gained his famous status The DISTANCE formula was "created" by by founding a group, the Brotherhood of Pythagoreans, which was devoted to manipulating the Pythagorean Theorem. the study of mathematics. The group was almost cult-like in that it had symbols, rituals and prayers. In addition, Pythagoras believed that "Number rules the universe,"and the Pythagoreans gave numerical values to many objects and ideas. These numerical values, in turn, were endowed with mystical and spiritual qualities. Nov 5­9:21 AM Nov 5­9:21 AM 9
  10. 10. Unit 4 Notebook 2012.notebook December 04, 2012 Legend has it that upon completion of his famous theorem, If we take an isosceles right triangle with legs of measure 1, the hypotenuse will Pythagoras sacrificed 100 oxen. Although he is credited with the discovery of measure sqrt 2. But this number cannot be expressed as a length that can be the famous theorem, it is not possible to tell if Pythagoras is the actual author. measured with a ruler divided into fractional parts, and that deeply disturbed the The Pythagoreans wrote many geometric proofs, but it is difficult to ascertain Pythagoreans, who believed that "All is number." They called these numbers who proved what, as the group wanted to keep their findings secret. "alogon," which means "unutterable." So shocked were the Pythagoreans by these Unfortunately, this vow of secrecy prevented an important mathematical idea numbers, they put to death a member who dared to mention their from being made public. The Pythagoreans had discovered irrational numbers! existence to the public. It would be 200 years later that the Greek mathematician Eudoxus developed a way to deal with these unutterable numbers... 1 1 Nov 5­9:21 AM Nov 5­9:21 AM The Pythagorean Theorem is used to find the length of a side of a RIGHT TRIANGLE. a 2 + b 2 = c2 a & b represent the length of the legs The Pythago rean Theore m also states: c represents the length of the hypotenuse the area of plus the are square A is equal to th a of square B ***The hypotenuse is the longest side of the e area of sq right triangle. uare C. Nov 5­9:21 AM Nov 5­9:21 AM Pythagorean Triples: Common lengths of sides in a right triangle. How do you tell if any given 3 sides I bet you know at least one of these triples.... would form a right triangle? Use the Pythagorean Theorem to check!!! (3,4,5) (16, 63, 65) Determine if the following lengths form ( 5, 12, 13) (20, 21, 29) a right triangle. ( 7, 24, 25) (28, 45, 53) 1. 24 2. 15 ( 8, 15, 17) (33, 56, 65) 10 14 ( 9, 40, 41) (36, 77, 85) 26 7 (11, 60, 61) (39, 80, 89) (12, 35, 37) (48, 55, 73) (13, 84, 85) (65, 72, 97) Nov 5­9:21 AM Nov 5­9:21 AM 10
  11. 11. Unit 4 Notebook 2012.notebook December 04, 2012 Find the missing side of the right triangle and round to the Find the missing side of the right triangle and leave your nearest tenth. answer in simplest radical form. 2 6 12 15 3 4 2 10 6 1 5 5 Nov 5­9:21 AM Nov 5­9:21 AM Answers to p. 8 #1- 12 1. NO 2. YES 3. YES Homework: 4. YES 5. NO 6. YES p. 8 #1- 12 7. 8.9 8. 6.7 9. 12.2 10. 7.6 11. 7.3 12. 6.3 Nov 5­9:21 AM Nov 5­9:21 AM QUIZ #7 The Quadratic Formula Ha ha .. one is .This funny, pretty CR #7 due right?? ? tomorrow Nov 5­9:21 AM Nov 5­9:22 AM 11
  12. 12. Unit 4 Notebook 2012.notebook December 04, 2012 Solve for x in the following 2 problems: Often, the simplest way to solve "ax2 + bx + c = 0" for the value of x is to factor the quadratic, set 1. x2 ­ 5x ­ 36 = 0 each factor equal to zero, and then solve each factor. 2. 2x2 + 2x = 12 But sometimes the quadratic is too messy, or it doesnt factor at all, or you just dont feel like factoring. While factoring may not always be successful, the Quadratic Formula can always find the solution. Nov 5­9:22 AM Nov 5­9:22 AM For ax2 + bx + c = 0, the value of x is given by: There are some songs that will help you memorize the Quadratic Formula. I use the song "Pop Goes The Weasel." http://www.youtube.com/watch?v=2lbABbfU6Zc This is a cute little video that students at Westerville South High School in Ohio created. Maybe you guys could make one??? This formula is called The Quadratic Formula. (Start at 3:00) http://www.youtube.com/watch?v=jGJrH49Z2ZA&feature=related ***Remember, a, b, and c represent the coefficients in our equation. Solving ax2 + bx + c = 0 for x means, among other things, that you are trying to find x­intercepts. This one is pretty bad... http://www.youtube.com/watch?v=TVIcjaKt_A8&feature=fvwrel Key Point: Sometimes they ask you to find the ROOTS. This is a fancy way of asking you to solve for the variable! Nov 5­9:22 AM Nov 5­9:22 AM Solve for x in each of the following in simplest radical Find the roots in each of the following equations in form:   simplest radical form:   Nov 5­9:22 AM Nov 5­9:22 AM 12
  13. 13. Unit 4 Notebook 2012.notebook December 04, 2012 ANSWERS to p. 10­ 11 #6, 8, and 16: 6.  x = ­4,  Homework:   p. 10, 11 #6, 8, and 16 8.  x =  16.  n =  Nov 5­9:22 AM Nov 5­9:22 AM Quadratic Formula with Remind me again how you would solve the following quadratic equation for the value of x in simplest radical form: Sum and Product Rules 1. Since this is an equation, there must be a way to check it. We could substitute our answer back into the original equation, but our answer is really messy. Plus there are 2 of them! There must be another method of check our roots of the quadratic equation. Nov 5­9:22 AM Nov 5­9:22 AM The method we will use to check our answers to a quadratic equation is called the SUM RULE and PRODUCT RULE Product Rule: Sum Rule: Nov 5­9:22 AM Nov 5­9:22 AM 13
  14. 14. Unit 4 Notebook 2012.notebook December 04, 2012 Review Nov 5­9:22 AM Nov 5­9:23 AM 14

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