4 2 rules of radicals

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4 2 rules of radicals

  1. 1. Rules of Radicals
  2. 2. Square Rule: x2 =x x = x if x > 0. Rules of Radicals
  3. 3. Square Rule: x2 =x x = x if x > 0. Multiplication Rule: x·y = x·y Rules of Radicals
  4. 4. Square Rule: x2 =x x = x if x > 0. Multiplication Rule: x·y = x·y We use these rules to simplify root-expressions. Rules of Radicals
  5. 5. Square Rule: x2 =x x = x if x > 0. Multiplication Rule: x·y = x·y We use these rules to simplify root-expressions. In particular, look for square factors of the radicand to pull out when simplifying square-root. Rules of Radicals
  6. 6. Example A. Simplify a. 8 Square Rule: x2 =x x = x if x > 0. Multiplication Rule: x·y = x·y We use these rules to simplify root-expressions. In particular, look for square factors of the radicand to pull out when simplifying square-root. Rules of Radicals
  7. 7. Example A. Simplify a. 8 = 42 Square Rule: x2 =x x = x if x > 0. Multiplication Rule: x·y = x·y We use these rules to simplify root-expressions. In particular, look for square factors of the radicand to pull out when simplifying square-root. Rules of Radicals
  8. 8. Example A. Simplify a. 8 = 42 = 22 Square Rule: x2 =x x = x if x > 0. Multiplication Rule: x·y = x·y We use these rules to simplify root-expressions. In particular, look for square factors of the radicand to pull out when simplifying square-root. Rules of Radicals
  9. 9. Example A. Simplify a. 8 = 42 = 22 Square Rule: x2 =x x = x if x > 0. Multiplication Rule: x·y = x·y We use these rules to simplify root-expressions. In particular, look for square factors of the radicand to pull out when simplifying square-root. Rules of Radicals b. 72 =
  10. 10. Example A. Simplify a. 8 = 42 = 22 Square Rule: x2 =x x = x if x > 0. Multiplication Rule: x·y = x·y We use these rules to simplify root-expressions. In particular, look for square factors of the radicand to pull out when simplifying square-root. Rules of Radicals b. 72 =362
  11. 11. Example A. Simplify a. 8 = 42 = 22 Square Rule: x2 =x x = x if x > 0. Multiplication Rule: x·y = x·y We use these rules to simplify root-expressions. In particular, look for square factors of the radicand to pull out when simplifying square-root. Rules of Radicals b. 72 =362 = 62
  12. 12. Example A. Simplify a. 8 = 42 = 22 Square Rule: x2 =x x = x if x > 0. Multiplication Rule: x·y = x·y We use these rules to simplify root-expressions. In particular, look for square factors of the radicand to pull out when simplifying square-root. Rules of Radicals b. 72 =362 = 62 c. x2y
  13. 13. Example A. Simplify a. 8 = 42 = 22 Square Rule: x2 =x x = x if x > 0. Multiplication Rule: x·y = x·y We use these rules to simplify root-expressions. In particular, look for square factors of the radicand to pull out when simplifying square-root. Rules of Radicals b. 72 =362 = 62 c. x2y =x2y
  14. 14. Example A. Simplify a. 8 = 42 = 22 Square Rule: x2 =x x = x if x > 0. Multiplication Rule: x·y = x·y We use these rules to simplify root-expressions. In particular, look for square factors of the radicand to pull out when simplifying square-root. Rules of Radicals b. 72 =362 = 62 c. x2y =x2y = xy
  15. 15. Example A. Simplify a. 8 = 42 = 22 Square Rule: x2 =x x = x if x > 0. Multiplication Rule: x·y = x·y We use these rules to simplify root-expressions. In particular, look for square factors of the radicand to pull out when simplifying square-root. Rules of Radicals b. 72 =362 = 62 d. x2y3 c. x2y =x2y = xy
  16. 16. Example A. Simplify a. 8 = 42 = 22 Square Rule: x2 =x x = x if x > 0. Multiplication Rule: x·y = x·y We use these rules to simplify root-expressions. In particular, look for square factors of the radicand to pull out when simplifying square-root. Rules of Radicals b. 72 =362 = 62 d. x2y3 =x2y2y c. x2y =x2y = xy
  17. 17. Example A. Simplify a. 8 = 42 = 22 Square Rule: x2 =x x = x if x > 0. Multiplication Rule: x·y = x·y We use these rules to simplify root-expressions. In particular, look for square factors of the radicand to pull out when simplifying square-root. Rules of Radicals b. 72 =362 = 62 d. x2y3 =x2y2y = xyy c. x2y =x2y = xy
  18. 18. Example A. Simplify a. 8 = 42 = 22 Square Rule: x2 =x x = x if x > 0. Multiplication Rule: x·y = x·y We use these rules to simplify root-expressions. In particular, look for square factors of the radicand to pull out when simplifying square-root. Rules of Radicals b. 72 =362 = 62 d. x2y3 =x2y2y = xyy c. x2y =x2y = xy A radical expression is said to be simplified if as much as possible is extracted out of the square-root.
  19. 19. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Rules of Radicals
  20. 20. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a. 72 Rules of Radicals
  21. 21. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a. 72 = 4 18 Rules of Radicals
  22. 22. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a. 72 = 4 18 = 218 (not simplified yet) Rules of Radicals
  23. 23. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a. 72 = 4 18 = 218 (not simplified yet) = 292 Rules of Radicals
  24. 24. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a. 72 = 4 18 = 218 (not simplified yet) = 292 = 2*3*2 Rules of Radicals
  25. 25. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a. 72 = 4 18 = 218 (not simplified yet) = 292 = 2*3*2 = 62 (simplified) Rules of Radicals
  26. 26. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a. 72 = 4 18 = 218 (not simplified yet) = 292 = 2*3*2 = 62 (simplified) b.80x4y5 Rules of Radicals
  27. 27. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a. 72 = 4 18 = 218 (not simplified yet) = 292 = 2*3*2 = 62 (simplified) b.80x4y5 = 16·5x4y4y Rules of Radicals
  28. 28. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a. 72 = 4 18 = 218 (not simplified yet) = 292 = 2*3*2 = 62 (simplified) b.80x4y5 = 16·5x4y4y = 4x2y25y Rules of Radicals
  29. 29. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a. 72 = 4 18 = 218 (not simplified yet) = 292 = 2*3*2 = 62 (simplified) b.80x4y5 = 16·5x4y4y = 4x2y25y Rules of Radicals Division Rule: y x y x  =
  30. 30. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a. 72 = 4 18 = 218 (not simplified yet) = 292 = 2*3*2 = 62 (simplified) b.80x4y5 = 16·5x4y4y = 4x2y25y Rules of Radicals Division Rule: y x y x  = Example C. Simplify.
  31. 31. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a. 72 = 4 18 = 218 (not simplified yet) = 292 = 2*3*2 = 62 (simplified) b.80x4y5 = 16·5x4y4y = 4x2y25y Rules of Radicals Division Rule: y x y x  = Example C. Simplify. 9 4 a.
  32. 32. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a. 72 = 4 18 = 218 (not simplified yet) = 292 = 2*3*2 = 62 (simplified) b.80x4y5 = 16·5x4y4y = 4x2y25y Rules of Radicals Division Rule: y x y x  = Example C. Simplify. 9 4 9 4 a. =
  33. 33. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a. 72 = 4 18 = 218 (not simplified yet) = 292 = 2*3*2 = 62 (simplified) b.80x4y5 = 16·5x4y4y = 4x2y25y Rules of Radicals Division Rule: y x y x  = Example C. Simplify. 9 4 9 4 3 2 a. = =
  34. 34. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a. 72 = 4 18 = 218 (not simplified yet) = 292 = 2*3*2 = 62 (simplified) b.80x4y5 = 16·5x4y4y = 4x2y25y Rules of Radicals Division Rule: y x y x  = Example C. Simplify. 9 4 9 4 3 2 9y2 x2 a. = = b.
  35. 35. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a. 72 = 4 18 = 218 (not simplified yet) = 292 = 2*3*2 = 62 (simplified) b.80x4y5 = 16·5x4y4y = 4x2y25y Rules of Radicals Division Rule: y x y x  = Example C. Simplify. 9 4 9 4 3 2 9y2 x2 9y2 x2 a. = = b. =
  36. 36. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a. 72 = 4 18 = 218 (not simplified yet) = 292 = 2*3*2 = 62 (simplified) b.80x4y5 = 16·5x4y4y = 4x2y25y Rules of Radicals Division Rule: y x y x  = Example C. Simplify. 9 4 9 4 3 2 9y2 x2 9y2 x2 3y x a. = = b. = =
  37. 37. The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, Rules of Radicals
  38. 38. The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. Rules of Radicals
  39. 39. The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals
  40. 40. Example D. Simplify 5 3 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals a. 
  41. 41. Example D. Simplify 5 3 5·5 3·5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals a. = 
  42. 42. Example D. Simplify 5 3 5·5 3·5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals a. =  = 25 15 
  43. 43. Example D. Simplify 5 3 5·5 3·5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals a. =  = 25 15  = 5 15
  44. 44. Example D. Simplify 5 3 5·5 3·5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals a. =  = 25 15  = 5 15 8x 5b.  5 1 15or
  45. 45. Example D. Simplify 5 3 5·5 3·5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals a. =  = 25 15  = 5 15 8x 5 4·2x 5b. =  5 1 15or
  46. 46. Example D. Simplify 5 3 5·5 3·5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals 2 a. =  = 25 15  = 5 15 8x 5 4·2x 5b. =  = 2x 5  5 1 15or
  47. 47. Example D. Simplify 5 3 5·5 3·5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals 2 a. =  = 25 15  = 5 15 8x 5 4·2x 5b. =  = 2x 5  = 2 2x 5  2x 2x  5 1 15or
  48. 48. Example D. Simplify 5 3 5·5 3·5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals 2 a. =  = 25 15  = 5 15 8x 5 4·2x 5b. =  = 2x 5  = 2 2x 5  2x 2x  = 2 2x 10x * 5 1 15or
  49. 49. Example D. Simplify 5 3 5·5 3·5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals 2 a. =  = 25 15  = 5 15 8x 5 4·2x 5b. =  = 2x 5  = 2 2x 5  2x 2x  = 2 2x 10x * = 4x 10x 5 1 15or
  50. 50. Example D. Simplify 5 3 5·5 3·5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals 2 a. =  = 25 15  = 5 15 8x 5 4·2x 5b. =  = 2x 5  = 2 2x 5  2x 2x  = 2 2x 10x * = 4x 10x 5 1 15or 4x 1 10xor
  51. 51. Example D. Simplify 5 3 5·5 3·5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals 2 a. =  = 25 15  = 5 15 8x 5 4·2x 5b. =  = 2x 5  = 2 2x 5  2x 2x  = 2 2x 10x * = 4x 10x WARNING!!!! a ± b = a ±b 5 1 15or 4x 1 10xor
  52. 52. Example D. Simplify 5 3 5·5 3·5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals 2 a. =  = 25 15  = 5 15 8x 5 4·2x 5b. =  = 2x 5  = 2 2x 5  2x 2x  = 2 2x 10x * = 4x 10x WARNING!!!! a ± b = a ±b For example: 4 + 913 = 5 1 15or 4x 1 10xor
  53. 53. Example D. Simplify 5 3 5·5 3·5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals 2 a. =  = 25 15  = 5 15 8x 5 4·2x 5b. =  = 2x 5  = 2 2x 5  2x 2x  = 2 2x 10x * = 4x 10x WARNING!!!! a ± b = a ±b For example: 4 + 913 = 5 1 15or 4x 1 10xor
  54. 54. Example D. Simplify 5 3 5·5 3·5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals 2 a. =  = 25 15  = 5 15 8x 5 4·2x 5b. =  = 2x 5  = 2 2x 5  2x 2x  = 2 2x 10x * = 4x 10x WARNING!!!! a ± b = a ±b For example: 4 + 9 = 4 +913 = 5 1 15or 4x 1 10xor
  55. 55. Example D. Simplify 5 3 5·5 3·5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals 2 a. =  = 25 15  = 5 15 8x 5 4·2x 5b. =  = 2x 5  = 2 2x 5  2x 2x  = 2 2x 10x * = 4x 10x WARNING!!!! a ± b = a ±b For example: 4 + 9 = 4 +9 = 2 + 3 = 513 = 5 1 15or 4x 1 10xor
  56. 56. Rules of Radicals Exercise A. Simplify the following radicals. 1. 12 2. 18 3. 20 4. 28 5. 32 6. 36 7. 40 8. 45 9. 54 10. 60 11. 72 12. 84 13. 90 14. 96x2 15. 108x3 16. 120x2y2 17. 150y4 18. 189x3y2 19. 240x5y8 18. 242x19y34 19. 12 12 20. 1818 21. 2 16 23. 183 22. 123 24. 1227 25. 1850 26. 1040 27. 20x15x 28.12xy15y 29. 32xy324x5 30. x8y13x15y9 Exercise B. Simplify the following radicals. Remember that you have a choice to simplify each of the radicals first then multiply, or multiply the radicals first then simplify.
  57. 57. Rules of Radicals Exercise C. Simplify the following radicals. Remember that you have a choice to simplify each of the radicals first then multiply, or multiply the radicals first then simplify. Make sure the denominators are radical–free. 8x 531. x 10  14 5x32. 7 20  5 1233. 15 8x 534. 3 2  3 32x35. 7 5  5 236. 29 x  x (x + 1)39. x (x + 1)  x (x + 1)40. x(x + 1) 1  1 (x + 1) 37. x (x2 – 1)41. x(x + 1) (x – 1)  x (x + 1)38. x21 – 1 Exercise D. Take the denominators of out of the radical. 42. 9x21 – 143.

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