WARM-UP Find the prime factorization for the following numbers. Write your answer as a product of primes. 1. 72 2. 120 5.1a Rational Exponents & Simplify Radicals Objective: To simplify rational exponents n n2 n3 n4 n5 n6 n7 n8 2 4 8 16 32 64 128 256 3 9 27 81 243 729 2187 4 16 64 256 1024 5 25 125 625 6 36 216 1269 7 49 343 2401 8 64 512 9 81 729 Radicals Root index n radicand x2 x4 x6 x8 x10 x even What should you do if the exponent is not even? x9 x even x1 x5 x15 Simplify: All variables are positive. 25x2 y 7 20x5 y10 Cube roots: Look for perfect cubes in the coefficient. How can you determine if the variable is a perfect cube? 3 27 x 6 y10 3 40x 7 y12z20 4 32x2 y 4z15 Rational Exponents n x the n th root of x 1 n x xn 23 = 8 54 = 625 General : 38 8 1 3 2 4 625 625 power a root n a x xn x 1 4 5 Now – Try some fun problems! 2 27 3 5 64 6 3 4 2 2 8 3 9 27 Remember: Root first makes the number smaller. 3 256 4 Can you simplify rational exponents? Assign 5.1a: 17-39 odd, 41-58 all 243 2 5 WARM-UP Hmmm…do you remember?? 1. x2x3 = 3. a-3 2. 4. (x2)3 = x x 5 2 = 5.1a Answers 42. 16 44. 27 46. 27 48. 1/3 50. 1/8 52. 7/4 54. 4/9 56. 15 58. 12/35 5.1b Simplifying Radical Expressions Objective: To simplify rational expressions using exponent properties Recall the exponent properties. a +b xaxb = x (xa)b = x ab a-n 1 an = xa b = x x a b x0 = 1, x 0 Now try these! 2 3 Ex.1 x x Ex. 2 1 2 x Ex.3 4 2 2 3 3 2 2 3 x Ex. 4 x Ex.5 x 2 x Ex.6 42 4 3 2 3 2 Simplify. All variables are positive. 6 4 Ex.10 8 x y z 1 2 2 9 3 Ex.7 3( x y ) 24 xy 2 Ex.8 8 3 yx 2 3 2 3 1 3 4 3 23 x Ex.9 2 4 x 4 3 Can you simplify rational expressions using exponent properties? Homework: 5.1: 59-67 odd, 68-78 all, 93, 94, 107, 108 Quiz after 5,3 5.1b Answers 68. 3 1 2 x 5 y10 z 3 11 70. b 3 4 a 15 1 72. y 74. a3b5 6 1 1 1 76. r32 s 2 78. b3 a 94. t 9.52, d 14,400 108. x3 4 x4 3 on the int erval 0 x 1 5.2 WARM-UP Simplify: 3 16x2 y8 1 2 27x9y12 1 3 5.2 More Rational Exponents Objective: To continue multiplying rational exponents Multiply: Ex1. 4x2 (3x3 2x 1) Ex2. 5 2 2x 3 4x 3 1 2x 2 1 2 2 3 3 Ex 3. x 3 x 4 1 1 2 2 Ex 4. x 5 x 5 3 2 3 2 Ex5. x 3 4 2 x 3 4 2 Multiply: 1 2 1 1 2 1 Ex6. x 3 y 3 x 3 x 3 y 3 y 3 Factor completely: Ex 7. 4(x – 3)2 + 5x(x – 3) = Factor with Rational Exponents Determine the smallest exponent and factor this from all terms. Ex8. x 5 x3 1 x2 2 x3 4 x5 8 x7 2 2x 3 6 1 x3 2 x5 4 x7 Ex9. 2 1 4x 5 y 8xy 2 Try these: 10 2x 3 5 5x 3 12 6 4 3x 3 2 x3 2 6x 5 1 13x 5 4 6 6 6 Last one! Add: Don’t forget the common denominator! 4 x x 4 1 x2 x x3 x 1 1 2 2x 1 1 2 Can you multiplying rational exponents? Assign 5.2: 3-69 (x3), 77, 81, 97-100 5.2 Answers 6. 20x3 15x 2 10x 1 1 40x 2 y 2 24. 25x 42. 1 5 6x 2 16y 1 15 5 7 4 t 7 4 x 60. 100. 1 k6 12. a 8a 30. 48. 66. 1 2 12 t - 125 1 4x 1 3 x 3 x7 x 1 4 18. t 6t 1 2 9 36. a + 27 1 1 3 54. x 2 x 3 3 98. x 3 2 5.3 Simplified Radical Form Objective: To write Radicals in simplest radical form Properties for radicals: a, b > 0 1. n a b n a n b na a 2. n n for b 0 b b 3. a b c d ac bd Simplify each radical means: No perfect squares left under the No perfect cubes left under the No fractions under the radical No radicals in the denominator 3 No factors in the radicand can be written as powers of the index. When you simplified radicals to this point the book said that all variables were positive. What if they do not tell us all variables are positive? x2 x : When you have an even root and an even exponent in the radicand that becomes an odd exponent when removed, you must use absolute value. x6 x3 The first one needs absolute value symbols to insure the answer is positive x7 x3 x The second does not because if x was negative, it could not be under an even root. Simplify each: Do not assume variables are positive. 3 18x10 y17z 4 8 4 x 4 y10z21 Type 1: Similar to section 5.1 Ex1. 2 4 162x7 y12z 41 Ex2. 3 48x3y 4z22 Type 2: No radicals in the denominator. 5 3 4x 3y 16 5 7 34 2 15 6 4 4 x 3 2 3y Try these: 4x 4 9xy 4 5 1 3 2x2 8xy 3 9x yz 2 5x2 y 7 3 2x8 8xy 10 How do you know what degree to make the exponents in the denominator? Can you write Radicals in simplest radical form? Assign 5.3: 3-21 (x3), 23-33 odd, 48-69 (x3), 71-77 odd, 85-87 all, 89, 105 GROUP ACTIVITY Learning Target: Find a set containing 3 equivalent forms of the same number on the face. You will work with the 1 or 2 people sitting beside you. Begin with all of the cards face-up spread out on the desk. Take turns gathering sets of 3 cards. 5.3 Answers 6. 8 2 12. 18 3 18. 8 33 48. 4 54. 135x 2 3x 5xy 6yz 66. 3z 60. 86. 6 6 3 10xy 2 2y 3 3 847 feet 22 Review 5-1 to 5-3 Questions? Remember NO CALCULATOR! 5.1: Simplify radicals and rational exponents Write radical expressions with rational exponents Evaluate rational exponents Simplify expressions with rational exponents 5.2: Multiply and factor using rational exponents Add by making common denominators with rational exponents 5.3: Simplify radicals if the variables may not be positive No fractions under the radical No radicals in the denominator Be able to do these for any root Now let’s try some problems! Write using rational exponents: 8 3 2 5 x x Simplify: 3 2 3 9 16 16 4 2 3 27 x2 y 9z14 1 a4 2 1 x3 y 4 1 x6y Does 200x 4 y10z15 a 1 : x2 3 9x 1 y2 3 2 4 2 1 2 y x y ? Multiply: 1 1 a 3 5 3a 3 1 2 5 2x 2x 1 3 3x 5 Factor : 2 x3 1 5x 3 2 4x 5 6 2 2xx 5 3 3 4x 1 x 53 4 16x 5 1 15 x 5 2 24x 5 9 Simplify : 4 12 12 3 y 4 2 3 6x 2 4x 4 3 x3 x 22 3x 2 1 4x 22 1 Assign: Review WS 5.4 – Addition and Subtraction of Radicals Objective: To add and subtract radicals We all know how to simplify an equation such as: 2x +3y – 5x = 3y – 3x The process for addition and subtraction of radicals is very similar. To do so you must have the same index and the same radicand. 5 2 3 5 2 2 3 2 3 5 Lets try some! **You may need to simplify first! Ex.1 3 18 5 12 27 Ex2. Ex.3 Ex.4 3 4x 3 3 4x x 3 48x3 y 5 3 6x 6 y 3 6y3 4 1 3 2 Ex.5 4 x 18 2 8 Can you add and subtract radicals? Homework: 5.4 54. 3 2 5.4 Answers Simplify. WARM-UP (7 3)(7 3) (5 3)(5 3) What did you notice about the above? These are called CONJUGATES! 5.5 Multiplication and Division of Radicals Objective: To multiply and divide radicals Recall the radical properties we learned earlier in the chapter. a b c d ac bd Ex.1 4 5 2 15 Ex.3 2 3 6 3 6 Then simplify if possible. Ex.2 5 2 8 4 3 Ex.4 x 3 x 3 Therefore factorable!!! Ex.5 x 1 3 2 Now for division. Don’t forget to rationalize the denominator!! Multiply the numerator & denominator by the conjugate of the denominator. Then FOIL. 4 Ex.6 1 3 5 2 Ex.7 4 2 Ex.8 Can you multiply and divide radicals? Homework: 5.5 3 x 2 5.5 Answers 6. 4200 12. 105 14 3 5 18. 50 10 21 36. x - 22 42. 24. 25a 20 ab 4b 30. 3 48. 2 x 2 y xy 54. a 2 ab b a b 7 7 6 60. 11 x 5x 2 66. x 7 x 7 x 7 x 14 x 49 2 4x 5.6a Equations with Radicals Objective: To solve basic radical equations Recall: 4x – 5 = 23 +5 +5 4x = 28 4 4 x =7 Locate the variable. Undo order of operations to isolate the variable. Procedure: How is 4 x 5 23 similar? Locate and isolate the radical. +5 +5 4x = 28 4 4 ( x )2 = ( 7 )2 How do you undo the radical? x = 49 Always check these answers. When you square, you may get extraneous roots. Squaring Property: If both sides of an equation are squared, the solutions to the original equation are also solutions to the new equation. *You must square the entire side. *You must check for extraneous(extra) roots. BASIC: Ex1 : 2 x 3 4 8 Ex3 : 4 3 x 3 Ex2 : 4 2x 3 12 Medium: Ex 4 : (x 3)2 ( x 3 )2 * Isolate the radical on one side. * Square both sides (the entire side- FOIL) * Solve the quadratic. (How?) x2 – 6x + 9 = x – 3 x2 – 7x + 12 = 0 (x – 4) (x – 3) = 0 x = 4, 3 Check both answers – one generally does not work. You try these: Ex5 : a 2 a 10 0 Ex7 : 3 x 5 3 2x 7 Ex6 : 3x 3 4 Can you solve basic radical equations? Assign: 5-6 to # 35 5.6b Solve: Warm - up t7 t5 5.6b More Solving Radical Equations Objective: To solve radical equations with radicals on both sides and identiry extaneous roots What happens when you have two radicals that you cannot combine? x 5 x 2 * Two different roots & something else *Isolate the more complicated radical on one side and square both sides. (The entire side.) * Isolate the radical that is left and square both sides again. You try this one: x 5 3 x 8 Graphing y x 0 1 4 y 0 1 2 What is the domain: What is the range: x How would each change affect the graph? Give the domain and range for each. Last Ex1 324:: One yyyy3: xyx x2332x 1 Domain: Range: Summary: How can you make the root open left? Upside down? Can you solve radical equations with radicals on both sides and identity extaneous roots? Assign: Rest of 5.6 5.6b Solutions 42. 48. 54. 56. 58. 60. 4 5, 13 12 x 125 10 10,000 The plume would be smaller if there was a current. 5.6c Solving Equations with Rational Exponents Objective: To solve equations with rational exponents and understand extraneous roots * To solve an equation with a rational exponent, you must first solve for the variable or parenthesis with the rational exponent. * You must undo the exponent, by taking it to a power that will cancel the exponent to a 1. 1 3 3 ( )3 3 ( ) x 4 x Ex1 : 4 3 x 8 4 x1 = 64 Ex2 : 4x 2 3 5 4 How do you know when you should use for your solution? When solving an equation and you must take an even root, you must use x = answer. You try these: Ex3 : 3x 1 3 4 1 26 Ex5 : x 6 x 8 0 Ex7 : 4 x3 13x 2 3 36 0 Ex 4 : 4x 2 5 36 Ex6 : x 3 x 40 Miscellaneous Completely factor: x2n – 5xn + 6 Now try: Ex: 2x4n + x2n - 6 Ex: x2n+1 - 5xn+1 + 6x Cancel: x 3n x2 x 3n 5 xn x xn 1 2n 5 Can you solve equations with rational exponents and understand extraneous roots? Assignment: Worksheet and begin test review. 5.6c Worksheet Solutions 1. 2. 3. 4. 5. 6. 7. 27 16 32 -32 64 1 64 1 16 8. 14 9. 7 3 10. 4 3 11. 12. 13. 14. 81 1024 63, -62 341 15. 15 2 81 32 35 & -29 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 25 27 & -64 49 & 25 25 32,768 & -32 9 & .25 x(2x n 3)( x n 2) 2 x 3 ( x 3)( x 3) x 1 (2n2n 1)( x 2n 5) x 2 n1 x 2 n1 x 2n x n6 5.7a Introduction to Complex Numbers Objective: To define imaginary and complex numbers and perform simple operations on each Complex Numbers (C) : a + bi Imaginary (Im): Real Numbers (R): the set of rational and irrational numbers. negative Rational (Q) : any number that can be written as a fraction Irrational (Ir) : non-repeating, non-terminating decimals What are imaginary numbers? Integers (I or Z): positive and negative Whole numbers: no fractions or decimals Whole (W) : {0, 1, 2, 3, …} a = real part b = imaginary part Square roots of negative numbers. no fractions, decimals or negatives. Natural (N) : counting numbers no decimals, fractions, negatives, 0 Symbol? i Value? 1 1 i A complex number is in the form of a + bi where a = real part and bi = imaginary part. A pure imaginary number only has the imaginary part, bi. **Always remove the negative from the radical first!** Ex1: 4 Ex2: 4 9 Ex3: Ex4: 2 8 Ex5: 9 25 Ex6: 27 5 5 What is the value of i 2 ? ( 1 )2 ( i ) 2 -1 = i2 When you get an i2 , always replace it with a -1. i9 = i10 = i11 = i12 = i = i i2 = -1 i3 = i4 = i5 = i6 = i7 = i8 = Ex1: i20 = Ex2: i30 = Ex3: i57 = Ex4: i101 = Ex5: i12 i25 i-3 = For 2 complex numbers to be equal, the real parts must be equal and the imaginary parts must be equal. Ex1: 3x + 2i = 6 + 8yi Ex2: 4x – 3 + 2i = 9 – 6yi Ex3: 5 – (4 + y)i = 2x + 3 – 6i Can you define imaginary and complex numbers and perform simple operations on each? Assign: 1-24 all 5.7a Solutions 2. 4. 6. 8. 10. 12. 14. 16. 18. 7i -9i 4i 3 5i 3 -i i -1 x = 1 y = -4 x = 2/3 y = -.5 20. 22. 24. x = -.5 y = -5/3 x = 11/4 y = -2 x = 2/5 y = -4 5.7b Operations on Complex Numbers Objective: To perform operations on complex numbers Add/subtract: Add real part to real part and imaginary part to imaginary part Compare to: Add: (4 – 3x) + (2x – 8) = -x - 4 (2 + 4i) + (6 – 9i) = Subtract: (5 – 3i) – (7 – 5i) = Multiply: Distribute or FOIL – all answers should be in standard complex form: a + bi Don’t forget i2 = -1 Ex1: 2i(3 – 4i) = Ex2: -4i(5 + 6i) = Ex3: (2 – 3i)(4 + 5i) = Ex4: (4 – 2i)2 = This is similar to rationalizing the denominator with radicals. Division: Type 1: 4 i 5 2i -or- 4 Recall: How did we rationalize the denominator? 2 3 Use the complex conjugate to divide complex numbers. a + bi Type 2: 4 3 2i -or- 12 8i 9 4i Can you perform operations on complex numbers? Assign: 5.7b: 25-77 odd 87-90 a - bi 5.7 b Solutions 88. i if n is even 90. x = 1 – i is a solution to the equation. 5.7c and Review Objective: To factor and simplify using complex numbers FOIL: (x - 3)(x + 3) -compare to- (x – 3i)(x + 3i) This means the following can be factored. How? 1. 4x2 - 25 2. 4x2 + 25 3. x2 + 4 4. 2x2 + 98 Just for fun. (And they make great essay questions.) *What are imaginary numbers? *What symbol is used to designate imaginary units? *What is the value of the imaginary unit? *Give the definition of a complex number? *What is the complex conjugate and when should it be used? Give an example. *What is a pure imaginary number? Ex1 : i2 i30 i5 Ex2 : i13 i21 i8 i50 Ex4: 4x – 3 + 2yi = x + 2y – 8i Ex7 : 2i 4 3i Ex10 : 5i2 3i2i Ex5: Ex3 : i5 i12 i22 i30 8 6 Ex8 : 4 5i 8i 5 6i Ex11 : Solve : x2 4 0 Ex6 : 4 i 3i Ex9 : 3 2i2 Can you factor and simplify using complex numbers? Assign Worksheet: Mini-Quiz Tomorrow!! Worksheet Answers 1. i 11 2. 10i 10. 17i 11. 8i 2 19. 20. 2 23i 41 3 4i 2 28. 0 29. 1 3. 2i 2 12. 6 21. 17-6i 30. 1 4. -20 13. -5 22. 50 31. x2 - 36 5. -8 14. 5i 2 23. 1 4i 3 32. x2 + 36 6. i 15. -8 + 6i 24. -3 + 4i 33. x2 - 4 7. 15i 16. 2/3 25. 1 + 21i 34. x2 + 4 8. -35 17. -4i 26. -45 + 30i 35. (x+3)(x-3) 27. 1 36. (x+3i)(x-3i) 9. -36 18. 8 4i 5 37. (x+7)(x-7) 38. (x+7i)(x-7i) Ch 5 Review Answers 3 5 6 1. 2 a b c 6b 4. 7 x2 3 y 2z 7. x 2x 7 7 10. 8 4 5 8. 3. 6 2 3 3 5. 80 3 2 2 3 2. x 3xyz 6. 6 2 15 x2 4 12y3 5 7 9. 7 2y 2 11. 6 3 3 2 6 3 2 13. 2x 3x 1 3 20 14. 24 3 4 15. 2 27 17. x 6 18. x 100 19. x 1, 8 22. x 6 23. x 517 24. x 28, 26 26. x 4 27. Ø 12. 3x 5 14 x 1 16. x 5 3 20. y 1 25. x 13 21. x 9 28. a. y x 30. a. y 2 x 29. a. y x 2 b. y x 2 b. y 21 x b. y x 3 c. y x 2 b a c a b b a