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Prime Factorization and Rational Exponents Warm-Up

This warm-up activity involves finding the prime factorization of numbers and simplifying expressions with rational exponents. It also covers simplifying radical expressions and writing radicals in simplest form.

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Prime Factorization and Rational Exponents Warm-Up

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  1. WARM-UP Find the prime factorization for the following numbers. Write your answer as a product of primes. 1. 72 2. 120

  2. 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

  3. Radicals Root index What should you do if the exponent is not even?

  4. Simplify: All variables are positive. Cube roots: Look for perfect cubes in the coefficient. How can you determine if the variable is a perfect cube?

  5. Rational Exponents 23 = 8 Now – Try some fun problems! Remember: Root first makes the number smaller. 54 = 625 Can you simplify rational exponents? Assign 5.1a: 17-39 odd, 41-58 all

  6. WARM-UP Hmmm…do you remember?? • x2x3 = 2. (x2)3 = 3. a-3 4. =

  7. 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

  8. xaxb = (xa)b = a-n = 1, x0 = 5.1b Simplifying Radical Expressions Objective: To simplify rational expressions using exponent properties Recall the exponent properties.

  9. Now try these!

  10. Simplify. All variables are positive. 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   

  11. 5.1b Answers 68. 70. 72. 74. 76. 78. 94. 108.

  12. 5.2 WARM-UP Simplify:

  13. 5.2More Rational Exponents Multiply: Objective: To continue multiplying rational exponents

  14. Multiply: Factor completely: Ex 7. 4(x – 3)2 + 5x(x – 3) =

  15. Factor with Rational Exponents Try these: Determine the smallest exponent and factor this from all terms.

  16. Last one! Add: Don’t forget the common denominator! Can you multiplying rational exponents? Assign 5.2:3-69 (x3), 77, 81, 97-100

  17. 5.2 Answers 6. 12. 18. 24. 30. t - 125 36. a + 27 42. 48. 54. 60. 66. 98. 100.

  18. Simplify each radical means: • No perfect squares left under the • No perfect cubes left under the No factors in the radicand can be written as powers of the index. • No fractions under the radical • No radicals in the denominator 5.3 Simplified Radical Form Objective: To write Radicals in simplest radical form Properties for radicals: a, b > 0

  19. 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? 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. The second does not because if x was negative, it could not be under an even root. The first one needs absolute value symbols to insure the answer is positive Simplify each: Do not assume variables are positive.

  20. Type 1: Similar to section 5.1 Type 2: No radicals in the denominator.

  21. Try these: 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

  22. 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.

  23. 5.3 Answers 6. 12. 18. 48. 54. 60. 66. 86.

  24. 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 Review 5-1 to 5-3 Questions? Remember NO CALCULATOR! Now let’s try some problems! 

  25. Write using rational exponents: Simplify:

  26. Multiply: Assign: Review WS

  27. 5.4 – Addition and Subtraction of RadicalsObjective: To add and subtract radicals = 3y – 3x We all know how to simplify an equation such as: 2x +3y – 5x The process for addition and subtraction of radicals is very similar. To do so you must have the same indexand the same radicand. Lets try some! **You may need to simplify first!

  28. Can youadd and subtract radicals? Homework: 5.4

  29. 5.4 Answers 54.

  30. WARM-UP Simplify. What did you notice about the above? These are called CONJUGATES!

  31. 5.5 Multiplication and Division of RadicalsObjective: To multiply and divide radicals Recall the radical properties we learned earlier in the chapter. Then simplify if possible. Therefore factorable!!!

  32. Now for division. Don’t forget to rationalize the denominator!! Multiply the numerator & denominator by the conjugate of the denominator. Then FOIL. Can you multiply and divide radicals? Homework: 5.5

  33. 5.5 Answers • 6. 4200 12. 18. • 24. • 30. 3 36. x - 22 42. • 48. 54. 60. • 66.

  34. 5.6a Equations with RadicalsObjective: To solve basic radical equations +5 +5 +5 +5 4 x = 28 4x = 28 4 4 4 4 x = 7 x = 7 Procedure: How is similar? Locate and isolate the radical. x = 49 Always check these answers. When you square, you may get extraneous roots. • Locate the variable. Recall: 4x – 5 = 23 • Undo order of operations to isolate the variable. ( )2 ( )2 How do you undo the radical?

  35. BASIC: 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.

  36. Medium: * Isolate the radical on one side. * Square both sides (the entire side- FOIL) * Solve the quadratic. (How?) You try these: ( )2 ( )2 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. Can you solve basic radical equations? Assign: 5-6 to # 35

  37. 5.6b Warm - up Solve:

  38. 5.6b More Solving Radical EquationsObjective: To solve radical equations with radicals on both sides and identiry extaneous roots * 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. What happens when you have two radicals that you cannot combine?

  39. You try this one:

  40. x y 0 0 1 1 4 2 Graphing What is the domain: What is the range:

  41. How would each change affect the graph? Give the domain and range for each. 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

  42. 5.6b Solutions 42. 4 48. 5, 13 54. 56. 10 58. 10,000 60. The plume would be smaller if there was a current.

  43. 5.6c Solving Equations with Rational ExponentsObjective: 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. ( )3( )3 x1 = 64

  44. 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:

  45. Miscellaneous Cancel: Completely factor: x2n – 5xn + 6 Now try: Ex: 2x4n + x2n - 6 Ex: x2n+1 - 5xn+1 + 6x Can you solve equations with rational exponents and understand extraneous roots? Assignment: Worksheet and begin test review.

  46. 5.6c Worksheet Solutions • 27 10. 19. 25 • 16 20. 27 & -64 • 32 11. 81 21. 49 & 25 • -32 12. 1024 22. 25 • 64 13. 63, -62 23. 32,768 & -32 6. 14. 341 24. 9 & .25 25. 7. 15. 26. 8. 14 16. 81 27. 17. 32 28. 9. 18. 35 & -29 29. 30. 31.

  47. 5.7a Introduction to Complex NumbersObjective: To define imaginary and complex numbers and perform simple operations on each a = real part b = imaginary part Complex Numbers(C) : a + bi Imaginary (Im): Real Numbers (R): the set of rational and irrational numbers. Rational (Q) :any number that can be written as a fraction Irrational (Ir): non-repeating, non-terminating decimals Integers(I or Z): positive and negative Whole numbers: no fractions or decimals Whole(W): {0, 1, 2, 3, …} no fractions, decimals or negatives. Natural(N): counting numbers no decimals, fractions, negatives, 0 What are imaginary numbers? Symbol? Value? Square roots of negative numbers. i

  48. Ex1: Ex2: Ex3: Ex4: Ex5: Ex6: 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!**

  49. i = i i2 = -1 i3 = i4 = What is the value of i 2 ? ( )2 ( )2 -1 = i2 When you get an i2 , always replace it with a -1. i9 = i10 = i11 = i12 = Ex1: i20 = Ex2: i30 = Ex3: i57 = Ex4: i101 = Ex5: i12 i25 i-3 = i5 = i6 = i7 = i8 =

  50. 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

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