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AA Section 8-5

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Products with Radicals


Annville-Cleona School District ™

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AA Section 8-5

  1. 1. Section 8-5 Products with Radicals
  2. 2. Warm-up 1. A rectangle has an area of 10 ft2. Its length is 20 ft. Find the width. 5 2. (3t ) = ?
  3. 3. Warm-up 1. A rectangle has an area of 10 ft2. Its length is 20 ft. Find the width. A = 10 ft2 w =? l = 20 ft 5 2. (3t ) = ?
  4. 4. Warm-up 1. A rectangle has an area of 10 ft2. Its length is 20 ft. Find the width. A = 10 ft2 w =? A = lw l = 20 ft 5 2. (3t ) = ?
  5. 5. Warm-up 1. A rectangle has an area of 10 ft2. Its length is 20 ft. Find the width. A = 10 ft2 w =? A = lw l = 20 ft 10 = w 20 5 2. (3t ) = ?
  6. 6. Warm-up 1. A rectangle has an area of 10 ft2. Its length is 20 ft. Find the width. A = 10 ft2 w =? A = lw l = 20 ft 10 = w 20 10 w= 20 5 2. (3t ) = ?
  7. 7. Warm-up 1. A rectangle has an area of 10 ft2. Its length is 20 ft. Find the width. A = 10 ft2 w =? A = lw l = 20 ft 10 = w 20 10 w= 20 • 20 20 5 2. (3t ) = ?
  8. 8. Warm-up 1. A rectangle has an area of 10 ft2. Its length is 20 ft. Find the width. A = 10 ft2 w =? A = lw l = 20 ft 10 = w 20 10 w= 20 10 20 • = 20 20 20 5 2. (3t ) = ?
  9. 9. Warm-up 1. A rectangle has an area of 10 ft2. Its length is 20 ft. Find the width. A = 10 ft2 w =? A = lw l = 20 ft 10 = w 20 10 w= 20 10 20 20 • = = 20 20 2 20 5 2. (3t ) = ?
  10. 10. Warm-up 1. A rectangle has an area of 10 ft2. Its length is 20 ft. Find the width. A = 10 ft2 w =? A = lw l = 20 ft 10 = w 20 10 w= 20 10 20 20 4•5 • = = = 20 20 2 2 20 5 2. (3t ) = ?
  11. 11. Warm-up 1. A rectangle has an area of 10 ft2. Its length is 20 ft. Find the width. A = 10 ft2 w =? A = lw l = 20 ft 10 = w 20 10 w= 20 10 20 20 4•5 25 • = = = = 20 20 2 2 2 20 5 2. (3t ) = ?
  12. 12. Warm-up 1. A rectangle has an area of 10 ft2. Its length is 20 ft. Find the width. A = 10 ft2 w =? A = lw l = 20 ft 10 = w 20 10 w= 20 10 20 20 4•5 25 • = = = = =5 20 20 2 2 2 20 5 2. (3t ) = ?
  13. 13. Warm-up 1. A rectangle has an area of 10 ft2. Its length is 20 ft. Find the width. A = 10 ft2 w =? A = lw l = 20 ft 10 = w 20 10 w= 20 10 20 20 4•5 25 • = = = = = 5 ft 20 20 2 2 2 20 5 2. (3t ) = ?
  14. 14. Warm-up 1. A rectangle has an area of 10 ft2. Its length is 20 ft. Find the width. A = 10 ft2 w =? A = lw l = 20 ft 10 = w 20 10 w= 20 10 20 20 4•5 25 • = = = = = 5 ft 20 20 2 2 2 20 5 2. (3t ) = ? 5 5 5 (3t ) = 3 t
  15. 15. Warm-up 1. A rectangle has an area of 10 ft2. Its length is 20 ft. Find the width. A = 10 ft2 w =? A = lw l = 20 ft 10 = w 20 10 w= 20 10 20 20 4•5 25 • = = = = = 5 ft 20 20 2 2 2 20 5 2. (3t ) = ? 5 5 5 5 (3t ) = 3 t = 243t
  16. 16. How do you simplify radicals?
  17. 17. m m m Recall: (xy ) = x y
  18. 18. m m m Recall: (xy ) = x y 1 1 1 So: (xy ) = x y n n n
  19. 19. m m m Recall: (xy ) = x y 1 1 1 So: (xy ) = x y n n n Thus: n n xy = x n y
  20. 20. m m m Recall: (xy ) = x y 1 1 1 So: (xy ) = x y n n n Thus: n n xy = x n y Now we can break up and simplify a radical by factoring the number inside!
  21. 21. Root of a Product Theorem For all nonnegative real numbers x and y, and for any integer n ≥ 2, 1 1 1 n n n (xy ) = x y or xy = x y n n n
  22. 22. Example 1 Find the product. 3 3 4 • 16
  23. 23. Example 1 Find the product. 3 3 4 • 16 3 = 4 •16
  24. 24. Example 1 Find the product. 3 3 4 • 16 3 = 4 •16 3 = 64
  25. 25. Example 1 Find the product. 3 3 4 • 16 3 = 4 •16 3 = 64 =4
  26. 26. Example 2 Assume t > 0. Write as a single radical. 7 7 4 2 2t • 7t
  27. 27. Example 2 Assume t > 0. Write as a single radical. 7 7 4 2 2t • 7t 7 4 2 = 2t • 7t
  28. 28. Example 2 Assume t > 0. Write as a single radical. 7 7 4 2 2t • 7t 7 4 2 = 2t • 7t 7 6 = 14t
  29. 29. Simplifying Radicals Break up the radical into perfect powers that match the given root
  30. 30. Example 3 Simplify 3 635 128a b c
  31. 31. Example 3 Simplify 3 635 128a b c 3 6332 = 64 • 2a b c c
  32. 32. Example 3 Simplify 3 635 128a b c 3 6332 = 64 • 2a b c c 3 3 633 2 = 64a b c • 2c
  33. 33. Example 3 Simplify 3 635 128a b c 3 6332 = 64 • 2a b c c 3 3 633 2 = 64a b c • 2c 3 2 2 = 4a bc 2c
  34. 34. Simplifying an n th Root When you factor the expression in the radical into perfect nth powers
  35. 35. Example 4 Simplify 83 4 81x y
  36. 36. Example 4 Simplify 83 4 81x y 4 8 3 4 = 81x • y
  37. 37. Example 4 Simplify 83 4 81x y 4 8 3 4 = 81x • y 24 3 y = 3x
  38. 38. Example 5 Simplify 57 3 54x y
  39. 39. Example 5 Simplify 57 3 54x y 3 26 3 = 27 • 2x x y y
  40. 40. Example 5 Simplify 57 3 54x y 3 26 3 = 27 • 2x x y y 36 2 3 3 = 27x y • 2x y
  41. 41. Example 5 Simplify 57 3 54x y 3 26 3 = 27 • 2x x y y 36 2 3 3 = 27x y • 2x y 23 2 2x y = 3xy
  42. 42. Mean:
  43. 43. Mean: Also known as the average
  44. 44. Mean: Also known as the average Add up all the values, then divide by total number of values
  45. 45. Mean: Also known as the average Add up all the values, then divide by total number of values Arithmetic mean
  46. 46. Mean: Also known as the average Add up all the values, then divide by total number of values Arithmetic mean (because we’re adding)
  47. 47. Mean: Also known as the average Add up all the values, then divide by total number of values Arithmetic mean (because we’re adding) Geometric Mean:
  48. 48. Mean: Also known as the average Add up all the values, then divide by total number of values Arithmetic mean (because we’re adding) Geometric Mean: Multiply all the values in a set of numbers then take the nth root, where n is the total number of values in the set
  49. 49. Example 6 Find the geometric mean of the integers 1 to 10 to the nearest hundredth
  50. 50. Example 6 Find the geometric mean of the integers 1 to 10 to the nearest hundredth 1• 2 • 3 • 4 • 5 • 6 • 7 • 8• 9 •10
  51. 51. Example 6 Find the geometric mean of the integers 1 to 10 to the nearest hundredth 1• 2 • 3 • 4 • 5 • 6 • 7 • 8• 9 •10 = 3628800
  52. 52. Example 6 Find the geometric mean of the integers 1 to 10 to the nearest hundredth 1• 2 • 3 • 4 • 5 • 6 • 7 • 8• 9 •10 = 3628800 10 3628800
  53. 53. Example 6 Find the geometric mean of the integers 1 to 10 to the nearest hundredth 1• 2 • 3 • 4 • 5 • 6 • 7 • 8• 9 •10 = 3628800 10 3628800 ≈ 4.53
  54. 54. Homework p. 503 #1 - 29 “While one person hesitates because he feels inferior, the other is busy making mistakes and becoming superior.” - Henry C. Link

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