4 1 radicals and pythagorean theorem

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4 1 radicals and pythagorean theorem

  1. 1. Radicals and Pythagorean Theorem Frank Ma © 2011
  2. 2. Radicals and Pythagorean Theorem We need a simple graphing calculator such as TI-83 plus or a scientific calculator that displace the input from here on.
  3. 3. Radicals and Pythagorean Theorem We need a simple graphing calculator such as TI-83 plus or a scientific calculator that displace the input from here on. You may use the calculators on your personal digital devices (but no software that does algebra is allowed during tests).
  4. 4. Radicals and Pythagorean Theorem We need a simple graphing calculator such as TI-83 plus or a scientific calculator that displace the input from here on. You may use the calculators on your personal digital devices (but no software that does algebra is allowed during tests). Square Root
  5. 5. “9 is the square of 3” may be rephrased backwards as “3 is the square root of 9”. Radicals and Pythagorean Theorem We need a simple graphing calculator such as TI-83 plus or a scientific calculator that displace the input from here on. You may use the calculators on your personal digital devices (but no software that does algebra is allowed during tests). Square Root
  6. 6. “9 is the square of 3” may be rephrased backwards as “3 is the square root of 9”. Radicals and Pythagorean Theorem Definition: If a is > 0, and a2 = x, then we say a is the square root of x. This is written as sqrt(x) = a, or x = a. We need a simple graphing calculator such as TI-83 plus or a scientific calculator that displace the input from here on. You may use the calculators on your personal digital devices (but no software that does algebra is allowed during tests). Square Root
  7. 7. “9 is the square of 3” may be rephrased backwards as “3 is the square root of 9”. Example A. a. Sqrt(16) = c. –3 = Radicals and Pythagorean Theorem Definition: If a is > 0, and a2 = x, then we say a is the square root of x. This is written as sqrt(x) = a, or x = a. b. 1/9 = d. 3 = We need a simple graphing calculator such as TI-83 plus or a scientific calculator that displace the input from here on. You may use the calculators on your personal digital devices (but no software that does algebra is allowed during tests). Square Root
  8. 8. “9 is the square of 3” may be rephrased backwards as “3 is the square root of 9”. Example A. a. Sqrt(16) = 4 c. –3 = Radicals and Pythagorean Theorem Definition: If a is > 0, and a2 = x, then we say a is the square root of x. This is written as sqrt(x) = a, or x = a. b. 1/9 = d. 3 = We need a simple graphing calculator such as TI-83 plus or a scientific calculator that displace the input from here on. You may use the calculators on your personal digital devices (but no software that does algebra is allowed during tests). Square Root
  9. 9. “9 is the square of 3” may be rephrased backwards as “3 is the square root of 9”. Example A. a. Sqrt(16) = 4 c. –3 = Radicals and Pythagorean Theorem Definition: If a is > 0, and a2 = x, then we say a is the square root of x. This is written as sqrt(x) = a, or x = a. b. 1/9 = 1/3 d. 3 = We need a simple graphing calculator such as TI-83 plus or a scientific calculator that displace the input from here on. You may use the calculators on your personal digital devices (but no software that does algebra is allowed during tests). Square Root
  10. 10. “9 is the square of 3” may be rephrased backwards as “3 is the square root of 9”. Example A. a. Sqrt(16) = 4 c. –3 = doesn’t exist Radicals and Pythagorean Theorem Definition: If a is > 0, and a2 = x, then we say a is the square root of x. This is written as sqrt(x) = a, or x = a. b. 1/9 = 1/3 d. 3 = We need a simple graphing calculator such as TI-83 plus or a scientific calculator that displace the input from here on. You may use the calculators on your personal digital devices (but no software that does algebra is allowed during tests). Square Root
  11. 11. “9 is the square of 3” may be rephrased backwards as “3 is the square root of 9”. Example A. a. Sqrt(16) = 4 c. –3 = doesn’t exist Radicals and Pythagorean Theorem Definition: If a is > 0, and a2 = x, then we say a is the square root of x. This is written as sqrt(x) = a, or x = a. b. 1/9 = 1/3 d. 3 = 1.732.. (calculator) We need a simple graphing calculator such as TI-83 plus or a scientific calculator that displace the input from here on. You may use the calculators on your personal digital devices (but no software that does algebra is allowed during tests). Square Root
  12. 12. “9 is the square of 3” may be rephrased backwards as “3 is the square root of 9”. Example A. a. Sqrt(16) = 4 c. –3 = doesn’t exist Radicals and Pythagorean Theorem Definition: If a is > 0, and a2 = x, then we say a is the square root of x. This is written as sqrt(x) = a, or x = a. b. 1/9 = 1/3 d. 3 = 1.732.. (calculator) Note that the square of both +3 and –3 is 9, but we designate sqrt(9) or 9 to be +3. We need a simple graphing calculator such as TI-83 plus or a scientific calculator that displace the input from here on. You may use the calculators on your personal digital devices (but no software that does algebra is allowed during tests). Square Root
  13. 13. “9 is the square of 3” may be rephrased backwards as “3 is the square root of 9”. Example A. a. Sqrt(16) = 4 c. –3 = doesn’t exist Radicals and Pythagorean Theorem Definition: If a is > 0, and a2 = x, then we say a is the square root of x. This is written as sqrt(x) = a, or x = a. b. 1/9 = 1/3 d. 3 = 1.732.. (calculator) Note that the square of both +3 and –3 is 9, but we designate sqrt(9) or 9 to be +3. We say “–3” is the “negative of the square root of 9”. We need a simple graphing calculator such as TI-83 plus or a scientific calculator that displace the input from here on. You may use the calculators on your personal digital devices (but no software that does algebra is allowed during tests). Square Root
  14. 14. 0 02 = 0 0 = 0 1 12 = 1 1 = 1 2 22 = 4 4 = 2 3 32 = 9 9 = 3 4 42 = 16 16 = 4 5 52 = 25 25 = 5 6 62 = 36 36 = 6 7 72 = 49 49 = 7 8 82 = 64 64 = 8 9 92 = 81 81 = 9 10 102 = 100 100 = 10 11 112 = 121 121 = 11 Radicals and Pythagorean Theorem Following are the square numbers and square-roots that one needs to memorize.
  15. 15. 0 02 = 0 0 = 0 1 12 = 1 1 = 1 2 22 = 4 4 = 2 3 32 = 9 9 = 3 4 42 = 16 16 = 4 5 52 = 25 25 = 5 6 62 = 36 36 = 6 7 72 = 49 49 = 7 8 82 = 64 64 = 8 9 92 = 81 81 = 9 10 102 = 100 100 = 10 11 112 = 121 121 = 11 Radicals and Pythagorean Theorem Following are the square numbers and square-roots that one needs to memorize. These numbers are special because many mathematics exercises utilize square numbers.
  16. 16. 0 02 = 0 0 = 0 1 12 = 1 1 = 1 2 22 = 4 4 = 2 3 32 = 9 9 = 3 4 42 = 16 16 = 4 5 52 = 25 25 = 5 6 62 = 36 36 = 6 7 72 = 49 49 = 7 8 82 = 64 64 = 8 9 92 = 81 81 = 9 10 102 = 100 100 = 10 11 112 = 121 121 = 11 We may estimate the sqrt of other small numbers using this table. Radicals and Pythagorean Theorem Following are the square numbers and square-roots that one needs to memorize. These numbers are special because many mathematics exercises utilize square numbers.
  17. 17. 0 02 = 0 0 = 0 1 12 = 1 1 = 1 2 22 = 4 4 = 2 3 32 = 9 9 = 3 4 42 = 16 16 = 4 5 52 = 25 25 = 5 6 62 = 36 36 = 6 7 72 = 49 49 = 7 8 82 = 64 64 = 8 9 92 = 81 81 = 9 10 102 = 100 100 = 10 11 112 = 121 121 = 11 We may estimate the sqrt of other small numbers using this table. For example, 25 < 30 < 36 Radicals and Pythagorean Theorem Following are the square numbers and square-roots that one needs to memorize. These numbers are special because many mathematics exercises utilize square numbers.
  18. 18. 0 02 = 0 0 = 0 1 12 = 1 1 = 1 2 22 = 4 4 = 2 3 32 = 9 9 = 3 4 42 = 16 16 = 4 5 52 = 25 25 = 5 6 62 = 36 36 = 6 7 72 = 49 49 = 7 8 82 = 64 64 = 8 9 92 = 81 81 = 9 10 102 = 100 100 = 10 11 112 = 121 121 = 11 We may estimate the sqrt of other small numbers using this table. For example, 25 < 30 < 36 hence 25 < 30 < 36 Radicals and Pythagorean Theorem Following are the square numbers and square-roots that one needs to memorize. These numbers are special because many mathematics exercises utilize square numbers.
  19. 19. 0 02 = 0 0 = 0 1 12 = 1 1 = 1 2 22 = 4 4 = 2 3 32 = 9 9 = 3 4 42 = 16 16 = 4 5 52 = 25 25 = 5 6 62 = 36 36 = 6 7 72 = 49 49 = 7 8 82 = 64 64 = 8 9 92 = 81 81 = 9 10 102 = 100 100 = 10 11 112 = 121 121 = 11 We may estimate the sqrt of other small numbers using this table. For example, 25 < 30 < 36 hence 25 < 30 < 36 or 5 < 30 < 6 Radicals and Pythagorean Theorem Following are the square numbers and square-roots that one needs to memorize. These numbers are special because many mathematics exercises utilize square numbers.
  20. 20. 0 02 = 0 0 = 0 1 12 = 1 1 = 1 2 22 = 4 4 = 2 3 32 = 9 9 = 3 4 42 = 16 16 = 4 5 52 = 25 25 = 5 6 62 = 36 36 = 6 7 72 = 49 49 = 7 8 82 = 64 64 = 8 9 92 = 81 81 = 9 10 102 = 100 100 = 10 11 112 = 121 121 = 11 We may estimate the sqrt of other small numbers using this table. For example, 25 < 30 < 36 hence 25 < 30 < 36 or 5 < 30 < 6 Since 30 is about half way between 25 and 36, Radicals and Pythagorean Theorem Following are the square numbers and square-roots that one needs to memorize. These numbers are special because many mathematics exercises utilize square numbers.
  21. 21. 0 02 = 0 0 = 0 1 12 = 1 1 = 1 2 22 = 4 4 = 2 3 32 = 9 9 = 3 4 42 = 16 16 = 4 5 52 = 25 25 = 5 6 62 = 36 36 = 6 7 72 = 49 49 = 7 8 82 = 64 64 = 8 9 92 = 81 81 = 9 10 102 = 100 100 = 10 11 112 = 121 121 = 11 We may estimate the sqrt of other small numbers using this table. For example, 25 < 30 < 36 hence 25 < 30 < 36 or 5 < 30 < 6 Since 30 is about half way between 25 and 36, so we estimate that 30 5.5. Radicals and Pythagorean Theorem Following are the square numbers and square-roots that one needs to memorize. These numbers are special because many mathematics exercises utilize square numbers.
  22. 22. 0 02 = 0 0 = 0 1 12 = 1 1 = 1 2 22 = 4 4 = 2 3 32 = 9 9 = 3 4 42 = 16 16 = 4 5 52 = 25 25 = 5 6 62 = 36 36 = 6 7 72 = 49 49 = 7 8 82 = 64 64 = 8 9 92 = 81 81 = 9 10 102 = 100 100 = 10 11 112 = 121 121 = 11 We may estimate the sqrt of other small numbers using this table. For example, 25 < 30 < 36 hence 25 < 30 < 36 or 5 < 30 < 6 Since 30 is about half way between 25 and 36, so we estimate that 30 5.5. In fact 30 5.47722…. Radicals and Pythagorean Theorem Following are the square numbers and square-roots that one needs to memorize. These numbers are special because many mathematics exercises utilize square numbers.
  23. 23. Equations of the form x2 = c has two answers: x = + c or – c if c>0. Radicals and Pythagorean Theorem
  24. 24. Equations of the form x2 = c has two answers: x = + c or – c if c>0. Radicals and Pythagorean Theorem Example B. Solve the following equations. a. x2 = 25
  25. 25. Equations of the form x2 = c has two answers: x = + c or – c if c>0. Radicals and Pythagorean Theorem Example B. Solve the following equations. a. x2 = 25 x = ± 25
  26. 26. Equations of the form x2 = c has two answers: x = + c or – c if c>0. Radicals and Pythagorean Theorem Example B. Solve the following equations. a. x2 = 25 x = ± 25 = ±5
  27. 27. Equations of the form x2 = c has two answers: x = + c or – c if c>0. Radicals and Pythagorean Theorem Example B. Solve the following equations. a. x2 = 25 x = ± 25 = ±5 b. x2 = –4
  28. 28. Equations of the form x2 = c has two answers: x = + c or – c if c>0. Radicals and Pythagorean Theorem Example B. Solve the following equations. a. x2 = 25 x = ± 25 = ±5 b. x2 = –4 Solution does not exist.
  29. 29. Equations of the form x2 = c has two answers: x = + c or – c if c>0. Radicals and Pythagorean Theorem Example B. Solve the following equations. a. x2 = 25 x = ± 25 = ±5 b. x2 = –4 Solution does not exist. c. x2 = 8
  30. 30. Equations of the form x2 = c has two answers: x = + c or – c if c>0. Radicals and Pythagorean Theorem Example B. Solve the following equations. a. x2 = 25 x = ± 25 = ±5 b. x2 = –4 Solution does not exist. c. x2 = 8 x = ± 8
  31. 31. Equations of the form x2 = c has two answers: x = + c or – c if c>0. Radicals and Pythagorean Theorem Example B. Solve the following equations. a. x2 = 25 x = ± 25 = ±5 b. x2 = –4 Solution does not exist. c. x2 = 8 x = ± 8 ±2.8284.. by calculator
  32. 32. Equations of the form x2 = c has two answers: x = + c or – c if c>0. Radicals and Pythagorean Theorem Example B. Solve the following equations. a. x2 = 25 x = ± 25 = ±5 b. x2 = –4 Solution does not exist. c. x2 = 8 x = ± 8 ±2.8284.. by calculator exact answer approximate answer
  33. 33. Equations of the form x2 = c has two answers: x = + c or – c if c>0. Radicals and Pythagorean Theorem Example B. Solve the following equations. a. x2 = 25 x = ± 25 = ±5 b. x2 = –4 Solution does not exist. c. x2 = 8 x = ± 8 ±2.8284.. by calculator exact answer approximate answer In geometry square roots show up in distance problems because of Pythagorean theorem.
  34. 34. A right triangle is a triangle with a right angle as one of its angle. Radicals and Pythagorean Theorem
  35. 35. A right triangle is a triangle with a right angle as one of its angles. The longest side C of a right triangle is called the hypotenuse, Radicals and Pythagorean Theorem hypotenuse C
  36. 36. A right triangle is a triangle with a right angle as one of its angles. The longest side C of a right triangle is called the hypotenuse, the two sides A and B forming the right angle are called the legs. Radicals and Pythagorean Theorem hypotenuse legs A B C
  37. 37. A right triangle is a triangle with a right angle as one of its angles. The longest side C of a right triangle is called the hypotenuse, the two sides A and B forming the right angle are called the legs. Pythagorean Theorem Given a right triangle with labeling as shown, then A2 + B2 = C2 Radicals and Pythagorean Theorem hypotenuse legs A B C
  38. 38. Radicals and Pythagorean Theorem There are two types of problems that use the Pythagorean Theorem to find the sides of the right triangles
  39. 39. Radicals and Pythagorean Theorem There are two types of problems that use the Pythagorean Theorem to find the sides of the right triangles-finding the hypotenuse versus finding the legs.
  40. 40. Example C. Find the missing side of the following right triangles. Find the exact answer and the approximate answer. Draw. a. a = 5, b = 12, c = ? Radicals and Pythagorean Theorem There are two types of problems that use the Pythagorean Theorem to find the sides of the right triangles-finding the hypotenuse versus finding the legs.
  41. 41. Example C. Find the missing side of the following right triangles. Find the exact answer and the approximate answer. Draw. a. a = 5, b = 12, c = ? Radicals and Pythagorean Theorem There are two types of problems that use the Pythagorean Theorem to find the sides of the right triangles-finding the hypotenuse versus finding the legs.
  42. 42. Example C. Find the missing side of the following right triangles. Find the exact answer and the approximate answer. Draw. a. a = 5, b = 12, c = ? Since it is a right triangle, 122 + 52 = c2 Radicals and Pythagorean Theorem There are two types of problems that use the Pythagorean Theorem to find the sides of the right triangles-finding the hypotenuse versus finding the legs.
  43. 43. Example C. Find the missing side of the following right triangles. Find the exact answer and the approximate answer. Draw. a. a = 5, b = 12, c = ? Since it is a right triangle, 122 + 52 = c2 144 + 25 = c2 Radicals and Pythagorean Theorem There are two types of problems that use the Pythagorean Theorem to find the sides of the right triangles-finding the hypotenuse versus finding the legs.
  44. 44. Example C. Find the missing side of the following right triangles. Find the exact answer and the approximate answer. Draw. a. a = 5, b = 12, c = ? Since it is a right triangle, 122 + 52 = c2 144 + 25 = c2 169 = c2 Radicals and Pythagorean Theorem There are two types of problems that use the Pythagorean Theorem to find the sides of the right triangles-finding the hypotenuse versus finding the legs.
  45. 45. Example C. Find the missing side of the following right triangles. Find the exact answer and the approximate answer. Draw. a. a = 5, b = 12, c = ? Since it is a right triangle, 122 + 52 = c2 144 + 25 = c2 169 = c2 So c = 169 = 13 Radicals and Pythagorean Theorem There are two types of problems that use the Pythagorean Theorem to find the sides of the right triangles-finding the hypotenuse versus finding the legs.
  46. 46. Example C. Find the missing side of the following right triangles. Find the exact answer and the approximate answer. Draw. a. a = 5, b = 12, c = ? Since it is a right triangle, 122 + 52 = c2 144 + 25 = c2 169 = c2 So c = 169 = 13 Since length can’t be negative, therefore c = 13. Radicals and Pythagorean Theorem There are two types of problems that use the Pythagorean Theorem to find the sides of the right triangles-finding the hypotenuse versus finding the legs.
  47. 47. b. a = 5, c = 12, b = ? Radicals and Pythagorean Theorem
  48. 48. b. a = 5, c = 12, b = ? Radicals and Pythagorean Theorem
  49. 49. b. a = 5, c = 12, b = ? Since it is a right triangle, b2 + 52 = 122 Radicals and Pythagorean Theorem
  50. 50. b. a = 5, c = 12, b = ? Since it is a right triangle, b2 + 52 = 122 b2 + 25 = 144 Radicals and Pythagorean Theorem
  51. 51. b. a = 5, c = 12, b = ? Since it is a right triangle, b2 + 52 = 122 b2 + 25 = 144 b2 = 144 – 25 Radicals and Pythagorean Theorem
  52. 52. b. a = 5, c = 12, b = ? Since it is a right triangle, b2 + 52 = 122 b2 + 25 = 144 b2 = 144 – 25 So b = 119 10.9. Radicals and Pythagorean Theorem
  53. 53. b. a = 5, c = 12, b = ? Since it is a right triangle, b2 + 52 = 122 b2 + 25 = 144 b2 = 144 – 25 So b = 119 10.9. But length can’t be negative, therefore b = 119 10.9 Radicals and Pythagorean Theorem
  54. 54. b. a = 5, c = 12, b = ? Since it is a right triangle, b2 + 52 = 122 b2 + 25 = 144 b2 = 144 – 25 So b = 119 10.9. But length can’t be negative, therefore b = 119 10.9 Radicals and Pythagorean Theorem The Distance Formula
  55. 55. b. a = 5, c = 12, b = ? Since it is a right triangle, b2 + 52 = 122 b2 + 25 = 144 b2 = 144 – 25 So b = 119 10.9. But length can’t be negative, therefore b = 119 10.9 Radicals and Pythagorean Theorem The Distance Formula
  56. 56. b. a = 5, c = 12, b = ? Since it is a right triangle, b2 + 52 = 122 b2 + 25 = 144 b2 = 144 – 25 So b = 119 10.9. But length can’t be negative, therefore b = 119 10.9 Radicals and Pythagorean Theorem Let (x1, y1) and (x2, y2) be two points, D = distance between them, The Distance Formula (x1, y1) (x2, y2) D
  57. 57. b. a = 5, c = 12, b = ? Since it is a right triangle, b2 + 52 = 122 b2 + 25 = 144 b2 = 144 – 25 So b = 119 10.9. But length can’t be negative, therefore b = 119 10.9 Radicals and Pythagorean Theorem Let (x1, y1) and (x2, y2) be two points, D = distance between them, then D2 = Δx2 + Δy2 The Distance Formula (x1, y1) (x2, y2) Δy Δx D
  58. 58. b. a = 5, c = 12, b = ? Since it is a right triangle, b2 + 52 = 122 b2 + 25 = 144 b2 = 144 – 25 So b = 119 10.9. But length can’t be negative, therefore b = 119 10.9 Radicals and Pythagorean Theorem Let (x1, y1) and (x2, y2) be two points, D = distance between them, then D2 = Δx2 + Δy2 where The Distance Formula (x1, y1) (x2, y2) Δx = x2 – x1 Δy Δx = x2 – x1 D
  59. 59. b. a = 5, c = 12, b = ? Since it is a right triangle, b2 + 52 = 122 b2 + 25 = 144 b2 = 144 – 25 So b = 119 10.9. But length can’t be negative, therefore b = 119 10.9 Radicals and Pythagorean Theorem Let (x1, y1) and (x2, y2) be two points, D = distance between them, then D2 = Δx2 + Δy2 where The Distance Formula (x1, y1) (x2, y2) Δx = x2 – x1 Δy = y2 – y1and Δy = y2 – y1 Δx = x2 – x1 D by the Pythagorean Theorem.
  60. 60. b. a = 5, c = 12, b = ? Since it is a right triangle, b2 + 52 = 122 b2 + 25 = 144 b2 = 144 – 25 So b = 119 10.9. But length can’t be negative, therefore b = 119 10.9 Radicals and Pythagorean Theorem Let (x1, y1) and (x2, y2) be two points, D = distance between them, then D2 = Δx2 + Δy2 where The Distance Formula (x1, y1) (x2, y2) Δx = x2 – x1 Δy = y2 – y1and Δy = y2 – y1 Δx = x2 – x1by the Pythagorean Theorem. Hence we’ve the Distant Formula: D = √ Δx2 + Δy2 D
  61. 61. b. a = 5, c = 12, b = ? Since it is a right triangle, b2 + 52 = 122 b2 + 25 = 144 b2 = 144 – 25 So b = 119 10.9. But length can’t be negative, therefore b = 119 10.9 Radicals and Pythagorean Theorem Let (x1, y1) and (x2, y2) be two points, D = distance between them, then D2 = Δx2 + Δy2 where The Distance Formula (x1, y1) (x2, y2) Δx = x2 – x1 Δy = y2 – y1and Δy = y2 – y1 Δx = x2 – x1by the Pythagorean Theorem. Hence we’ve the Distant Formula: D = √ Δx2 + Δy2 = √ (x2 – x1)2 + (y2 – y1)2 D
  62. 62. Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3) – ( 2, -4) D Radicals and Pythagorean Theorem –3, 7 Δx Δy
  63. 63. Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3) – ( 2, -4) D Radicals and Pythagorean Theorem –3, 7 Δx Δy
  64. 64. Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3) – ( 2, -4) D Radicals and Pythagorean Theorem –3, 7 Δx Δy
  65. 65. Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3) – ( 2, -4) D Radicals and Pythagorean Theorem –3, 7 Δx Δy
  66. 66. Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3) – ( 2, -4) D 7 –3 Radicals and Pythagorean Theorem –3, 7 Δx Δy
  67. 67. Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3) – ( 2, -4) D = (–3)2 + 72 = 58 7.62 D 7 –3 Radicals and Pythagorean Theorem –3, 7 Δx Δy
  68. 68. Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3) – ( 2, -4) D = (–3)2 + 72 = 58 7.62 D 7 –3 Radicals and Pythagorean Theorem –3, 7 Δx Δy Higher Root
  69. 69. Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3) – ( 2, -4) D = (–3)2 + 72 = 58 7.62 D 7 –3 Radicals and Pythagorean Theorem –3, 7 Δx Δy Higher Root If r3 = x, then we say a is the cube root of x.
  70. 70. Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3) – ( 2, -4) D = (–3)2 + 72 = 58 7.62 D 7 –3 Radicals and Pythagorean Theorem –3, 7 Δx Δy Higher Root If r3 = x, then we say a is the cube root of x. We write this as x = r.3
  71. 71. Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3) – ( 2, -4) D = (–3)2 + 72 = 58 7.62 D 7 –3 Radicals and Pythagorean Theorem –3, 7 Δx Δy Higher Root If r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x,3
  72. 72. Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3) – ( 2, -4) D = (–3)2 + 72 = 58 7.62 D 7 –3 Radicals and Pythagorean Theorem –3, 7 Δx Δy Higher Root If r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x. 3 k
  73. 73. Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3) – ( 2, -4) D = (–3)2 + 72 = 58 7.62 D 7 –3 Radicals and Pythagorean Theorem –3, 7 Δx Δy Higher Root If r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x. In the cases of even roots, i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0. 3 k k
  74. 74. Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3) – ( 2, -4) D = (–3)2 + 72 = 58 7.62 D 7 –3 Radicals and Pythagorean Theorem –3, 7 Δx Δy 3 k k Example E. a. 8 = 3 Higher Root If r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x. In the cases of even roots, i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.
  75. 75. Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3) – ( 2, -4) D = (–3)2 + 72 = 58 7.62 D 7 –3 Radicals and Pythagorean Theorem –3, 7 Δx Δy 3 k k Example E. a. 8 = 2 3 Higher Root If r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x. In the cases of even roots, i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.
  76. 76. Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3) – ( 2, -4) D = (–3)2 + 72 = 58 7.62 D 7 –3 Radicals and Pythagorean Theorem –3, 7 Δx Δy 3 k k Example E. a. 8 = 2 b. –1 = 3 3 Higher Root If r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x. In the cases of even roots, i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.
  77. 77. Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3) – ( 2, -4) D = (–3)2 + 72 = 58 7.62 D 7 –3 Radicals and Pythagorean Theorem –3, 7 Δx Δy 3 k k Example E. a. 8 = 2 b. –1 = –1 3 3 Higher Root If r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x. In the cases of even roots, i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.
  78. 78. Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3) – ( 2, -4) D = (–3)2 + 72 = 58 7.62 D 7 –3 Radicals and Pythagorean Theorem –3, 7 Δx Δy 3 k k Example E. a. 8 = 2 b. –1 = –1 c. –27 = 3 3 3 Higher Root If r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x. In the cases of even roots, i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.
  79. 79. Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3) – ( 2, -4) D = (–3)2 + 72 = 58 7.62 D 7 –3 Radicals and Pythagorean Theorem –3, 7 Δx Δy 3 k k Example E. a. 8 = 2 b. –1 = –1 c. –27 = –3 3 3 3 Higher Root If r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x. In the cases of even roots, i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.
  80. 80. Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3) – ( 2, -4) D = (–3)2 + 72 = 58 7.62 D 7 –3 Radicals and Pythagorean Theorem –3, 7 Δx Δy 3 k k Example E. a. 8 = 2 b. –1 = –1 c. –27 = –3 d. 16 = 3 3 3 Higher Root If r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x. In the cases of even roots, i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0. 4
  81. 81. Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3) – ( 2, -4) D = (–3)2 + 72 = 58 7.62 D 7 –3 Radicals and Pythagorean Theorem –3, 7 Δx Δy 3 k k Example E. a. 8 = 2 b. –1 = –1 c. –27 = –3 d. 16 = 2 3 3 3 Higher Root If r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x. In the cases of even roots, i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0. 4
  82. 82. Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3) – ( 2, -4) D = (–3)2 + 72 = 58 7.62 D 7 –3 Radicals and Pythagorean Theorem –3, 7 Δx Δy 3 k k Example E. a. 8 = 2 b. –1 = –1 c. –27 = –3 d. 16 = 2 e. –16 = 3 3 3 Higher Root If r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x. In the cases of even roots, i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0. 4 4
  83. 83. Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3) – ( 2, -4) D = (–3)2 + 72 = 58 7.62 D 7 –3 Radicals and Pythagorean Theorem –3, 7 Δx Δy 3 k k Example E. a. 8 = 2 b. –1 = –1 c. –27 = –3 d. 16 = 2 e. –16 = not real 3 3 3 Higher Root If r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x. In the cases of even roots, i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0. 4 4
  84. 84. Example D. Find the distance between (–1, 3) and (2, –4). (-1, 3) – ( 2, -4) D = (–3)2 + 72 = 58 7.62 D 7 –3 Radicals and Pythagorean Theorem –3, 7 Δx Δy 3 k k Example E. a. 8 = 2 b. –1 = –1 c. –27 = –3 d. 16 = 2 e. –16 = not real f. 10 ≈ 2.15.. 3 3 3 Higher Root If r3 = x, then we say a is the cube root of x. We write this as x = r. In general, if r k = x, then we say r is the k’th root of x, and we write it as a = x. In the cases of even roots, i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0. 4 4 3
  85. 85. Radicals and Pythagorean Theorem Rational and Irrational Numbers
  86. 86. The number 2 is the length of the hypotenuse of the right triangle as shown. Radicals and Pythagorean Theorem Rational and Irrational Numbers 2 1 1
  87. 87. The number 2 is the length of the hypotenuse of the right triangle as shown. Radicals and Pythagorean Theorem Rational and Irrational Numbers 2 1 1 It can be shown that 2 can not be represented as a ratio of whole numbers i.e. P/Q, where P and Q are integers.
  88. 88. The number 2 is the length of the hypotenuse of the right triangle as shown. Radicals and Pythagorean Theorem Rational and Irrational Numbers 2 1 1 It can be shown that 2 can not be represented as a ratio of whole numbers i.e. P/Q, where P and Q are integers. Hence these numbers are called irrational (non–ratio) numbers.
  89. 89. The number 2 is the length of the hypotenuse of the right triangle as shown. Radicals and Pythagorean Theorem Rational and Irrational Numbers 2 1 1 It can be shown that 2 can not be represented as a ratio of whole numbers i.e. P/Q, where P and Q are integers. Hence these numbers are called irrational (non–ratio) numbers. Most real numbers are irrational, not fractions, i.e. they can’t be represented as ratios of two integers.
  90. 90. The number 2 is the length of the hypotenuse of the right triangle as shown. Radicals and Pythagorean Theorem Rational and Irrational Numbers 2 1 1 It can be shown that 2 can not be represented as a ratio of whole numbers i.e. P/Q, where P and Q are integers. Hence these numbers are called irrational (non–ratio) numbers. Most real numbers are irrational, not fractions, i.e. they can’t be represented as ratios of two integers. The real line is populated sparsely by fractional locations.
  91. 91. The number 2 is the length of the hypotenuse of the right triangle as shown. Radicals and Pythagorean Theorem Rational and Irrational Numbers 2 1 1 It can be shown that 2 can not be represented as a ratio of whole numbers i.e. P/Q, where P and Q are integers. Hence these numbers are called irrational (non–ratio) numbers. Most real numbers are irrational, not fractions, i.e. they can’t be represented as ratios of two integers. The real line is populated sparsely by fractional locations. The Pythagorean school of the ancient Greeks had believed that all the measurable quantities in the universe are fractional quantities. The “discovery” of these extra irrational numbers caused a profound intellectual crisis.
  92. 92. The number 2 is the length of the hypotenuse of the right triangle as shown. Radicals and Pythagorean Theorem Rational and Irrational Numbers 2 1 1 It can be shown that 2 can not be represented as a ratio of whole numbers i.e. P/Q, where P and Q are integers. Hence these numbers are called irrational (non–ratio) numbers. Most real numbers are irrational, not fractions, i.e. they can’t be represented as ratios of two integers. The real line is populated sparsely by fractional locations. The Pythagorean school of the ancient Greeks had believed that all the measurable quantities in the universe are fractional quantities. The “discovery” of these extra irrational numbers caused a profound intellectual crisis. It wasn’t until the last two centuries that mathematicians clarified the strange questions “How many and what kind of numbers are there?”
  93. 93. Radicals and Pythagorean Theorem Exercise A. Solve for x. Give both the exact and approximate answers. If the answer does not exist, state so. 1. x2 = 1 2. x2 – 5 = 4 3. x2 + 5 = 4 4. 2x2 = 31 5. 4x2 – 5 = 4 6. 5 = 3x2 + 1 7. 4x2 = 1 8. x2 – 32 = 42 9. x2 + 62 = 102 10. 2x2 + 7 = 11 11. 2x2 – 5 = 6 12. 4 = 3x2 + 5 x 3 4 Exercise B. Solve for x. Give both the exact and approximate answers. If the answer does not exist, state so. 13. 4 3 x14. x 12 515. x 1 116. 2 1 x17. 3 2 3 x18.
  94. 94. Radicals and Pythagorean Theorem x 4 19. x x20. 3 / 3 21. 4 3 5 2 6 / 3 Exercise C. Given the following information, find the rise and run from A to B i.e. Δx and Δy. Find the distance from A to B. A 22. B A 23. B 24. A = (2, –3) , B = (1, 5) 25. A = (1, 5) , B = (2, –3) 26. A = (–2 , –5) , B = (3, –2) 27. A = (–4 , –1) , B = (2, –3) 28. Why is the distance from A to B the same as from B to A?
  95. 95. Exercise D. Find the exact answer. Radicals and Pythagorean Theorem 3 –129. 30. 13 –12534. 8 31. –13 8 3 –2732. 33. –13 64 3 10035. 100036. 3 10,00037. 1,000,00038. 3 0.0139. 0.00140. 3 0.000141. 0.00000142. 3

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