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# Hprec5.1

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### Hprec5.1

1. 1. 5-1: Radicals & Rational Exponents © 2007 Roy L. Gover (www.mrgover.com) Learning Goals: •Define and apply rational exponents •Perform arithmetic operations with radicals • Simplify expressions with radicals or rational exponents
2. 2. Definition A rational number is any number that can be written in the form: , 0 m n n ≠
3. 3. Definition A rational exponent is any number that can be written in the form: , 0 m n b n ≠
4. 4. Definition m n mn b b= Rational exponent form Radical form
5. 5. Definition m n b b
6. 6. Definition n m b n is the index of the radical. If the index is left blank, it is understood to be 2. m is the exponent of the radical.
7. 7. Example Evaluate without using a calculator: 1 2 81 2 3 27 −
8. 8. Try This Evaluate without using a calculator: 1 29 9 3= ± 2 38 − 3 2 1 1 48 =
9. 9. Try This Evaluate: 1 1 2 2 64 64 − − 7 63 7 8 8 =
10. 10. Example Express using rational exponents: 3 2 x 2 4 84 16a b c
11. 11. Try This Express using rational exponents: 5 3 a 3 5 a
12. 12. Try This Express using rational exponents: 3 2 63 8x y z 2 23 2xy z
13. 13. Example Express using radicals: 2 5 x 2 1 3 3 a b 11 3 82 4 16 x y ( ) 11 10 2 102x y a
14. 14. Try This Express using radicals: 3 205 15x y 1 3 45 5 15 x y
15. 15. Try This Express using radicals: 6 10 16s t 3 5 4s t
16. 16. Definition Perfect Squares:the number or expression resulting from multiplying any whole number or expression by itself.
17. 17. Examples perfect square because: •1 •4 •9 •16 1 x 1=1 2 x 2=4 3 x 3 =9 4 x 4=16
18. 18. Try This On a sheet of paper,list all the perfect squares from 16 to 625.
19. 19. Solution 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400 441 484 529 576 625
20. 20. Example perfect square because: 2 x 4 x 6 y 8 ( )xy 2 x x x=g 2 2 4 x x x=g 3 3 6 y y y=g 4 4 8 ( ) ( ) ( )xy xy xy=g Do you see a pattern?
21. 21. Example Simplify by removing all perfect square factors: 8 12 2 y 5 200xy
22. 22. Try This Simplify by removing all perfect square factors: 200 10 2
23. 23. Try This Simplify by removing all perfect square factors: 48 4 3
24. 24. Try This Simplify by removing all perfect square factors: 28 2 7
25. 25. Try This Simplify by removing all perfect square factors: 63 3 7
26. 26. Try This Simplify by removing all perfect square factors: 5 2 5 7 a b c 5 2 2 bc a c
27. 27. Try This Simplify by removing all perfect square factors: 5 12x 2 2 3x x
28. 28. Try This Simplify by removing all perfect square factors: 4 3 x y 2 x y y
29. 29. Definition Radical multiplication: a b ab• = ab a b= • and
30. 30. Example 2 3 =g 6 = x y z = 6 x =
31. 31. Try This 2 6 2x =g 2 12x Can you write this answer in simpler form...
32. 32. Definition Radical Division: a a bb = a a b b = and
33. 33. Example 6 3 = 12 =
34. 34. Definition Radical Addition & Subtraction: ( )a x b x a b x+ = + ( )a x b x a b x− = − ( )a x b y a b xy+ ≠ +
35. 35. Example 2 x x+ = 5 7 3 7− = 7 3p q+ =
36. 36. Important Idea x The radicand In order to add or subtract radicals, the radicands must be the same
37. 37. Try This 5 3xy xy+ = 8 xy
38. 38. Try This 5 3xy ab+ = 5 3xy ab+ The radicands are not the same, therefore the terms cannot be combined.
39. 39. Example 6 54+ Add: 2 2 3 8+
40. 40. Important Idea Sometimes you can add unlike radicands after removing perfect square factors
41. 41. Try This Add: 2 3 48− 2 3−
42. 42. Definition Rationalizing the Denominator: A procedure for writing an equivalent expression without any radicals in the denominator.
43. 43. Example Rationalize the Denominator: 1 3
44. 44. Example Rationalize the Denominator: 3 5
45. 45. Try This Rationalize the Denominator: 2 11 5 2 55 5
46. 46. Example Rationalize the Denominator: 2 3 x y
47. 47. Try This Rationalize the Denominator: 3 2 2 x y 3 2 2 xy y
48. 48. Example Rationalize the Denominator: 2 2 x y+
49. 49. Try This Rationalize the Denominator: x y x y + − ( ) 2 x y x y + − 2x xy y x y + + − or
50. 50. Definition Simplified Radical Form: •No perfect square factors in the radicand •No fractional radicands •No radicals in denominator
51. 51. Important Idea On tests and homework, you are expected to leave answers in simplified radical form where appropriate.
52. 52. Try This Re-write using simplified radical form: 1 4 1 2
53. 53. Try This Re-write using simplified radical form: 1 4 1 2
54. 54. Try This Re-write using simplified radical form: 3 12x 2 3x x
55. 55. Try This Re-write using simplified radical form: 4 2 xy − 4 2 14 xy xy+
56. 56. Properties of Exponents Product Property m n m n a a a + =
57. 57. Properties of Exponents Power of a Power Property ( )m n mn a a=
58. 58. Properties of Exponents Power of a Quotient Property , 0 m m m a a b b b   = ≠ ÷  
59. 59. Properties of Exponents Power of a Product Property ( )m m m ab a b=
60. 60. Properties of Exponents Quotient Property , 0 m m n n a a a a − = ≠
61. 61. Properties of Exponents Identity Property 1 a a=
62. 62. Properties of Exponents Reciprocal Property 1m m a a − = Do not leave answers with negative exponents
63. 63. Properties of Exponents 0 1a =
64. 64. Try This Evaluate 3 2 6 6 3 2 5 6 6+ = 1 1 3 3 − = 2 3 3 3
65. 65. Try This Evaluate ( ) 32 2 2 3 2 64x = 2 4 4 2 4x x=( ) 22 2x ( ) 04 2 13 3x x− − 1
66. 66. Example Simplify using only positive exponents: 2 2 ((2 ) )x −
67. 67. Try This Simplify using only positive exponents: 2 2 3 (2 )x y− 6 6 8x y
68. 68. Example 2 3 3 348r s−    ÷  ÷   Simplify using only positive exponents:
69. 69. Example Simplify using only positive exponents: 1 3 3 2 4 2x x x   − ÷  ÷  
70. 70. Example Simplify using only positive exponents: 25 7 42 4x y xy −     ÷ ÷  ÷ ÷   
71. 71. Try This Simplify using only positive exponents: 5 2 4 4516u v−    ÷  ÷  1 2 5 32u v Can you think of another way to write the answer?
72. 72. Try This Simplify using only positive exponents: Can you think of another way to write the answer? 27 5 33 6x y xy −     ÷ ÷  ÷ ÷    1 4 3 3x y
73. 73. Lesson Close Radical and exponential expressions are important for the mathematical description of a variety of problems in the physical sciences and business.
74. 74. Practice 1. pws5.1a (slides 1-15) 2. pws5.1b prob. 1-15 (slides 16-41) 3. pws5.1b prob. 16-31 (slides 42-55) 4. 334/55,56,63-72 all (slides 56-74)