SECTION 8-4
Radical Notation for nth Roots
WARM-UP
What is the length of the side of a cube whose volume
                      is 512 cm3?
WARM-UP
What is the length of the side of a cube whose volume
                      is 512 cm3?
WARM-UP
What is the length of the side of a cube whose volume
                      is 512 cm3?

                      V =...
WARM-UP
What is the length of the side of a cube whose volume
                      is 512 cm3?

                      V =...
WARM-UP
What is the length of the side of a cube whose volume
                      is 512 cm3?

                      V =...
WARM-UP
What is the length of the side of a cube whose volume
                      is 512 cm3?

                      V =...
WARM-UP
What is the length of the side of a cube whose volume
                      is 512 cm3?

                      V =...
WARM-UP
What is the length of the side of a cube whose volume
                      is 512 cm3?

                      V =...
n th   ROOTS
 1
x is the nth root of x
 n
n th   ROOTS
    1
  x is the nth root of x
    n




Square root of x:
n th   ROOTS
    1
  x is the nth root of x
    n




                        1
                    x
Square root of x:   ...
n th   ROOTS
    1
  x is the nth root of x
    n




                    1
                    x= x
Square root of x:   2
n th   ROOTS
    1
  x is the nth root of x
    n




                    1
                    x= x
Square root of x:   2...
n th   ROOTS
    1
  x is the nth root of x
    n




                        1
                    x= x
Square root of x:...
n th   ROOTS
    1
  x is the nth root of x
    n




                    1
                    x= x
Square root of x:   2...
n th   ROOTS
    1
  x is the nth root of x
    n




                    1
                    x= x
Square root of x:   2...
n th   ROOTS
    1
  x is the nth root of x
    n




                          1
                      x= x
Square root o...
n th     ROOTS
    1
  x is the nth root of x
    n




                        1
                     x= x
Square root of...
n th     ROOTS
                    1
                   x is the nth root of x
                    n




                 ...
DEFINITION



For any nonnegative real number x and any integer n ≥ 2,
                                 1
                ...
DEFINITION



For any nonnegative real number x and any integer n ≥ 2,
                                 1
                ...
EXAMPLE 1
Evaluate without using a calculator.



     3                         6
a.       27                 b. 64
EXAMPLE 1
Evaluate without using a calculator.



     3                         6
a.       27                 b. 64

 =3
EXAMPLE 1
Evaluate without using a calculator.



     3                         6
a.       27                 b. 64

 =3 ...
EXAMPLE 2
Use a calculator to estimate to the nearest hundredth.

                          5                    4
 a. 15 ...
EXAMPLE 2
Use a calculator to estimate to the nearest hundredth.

                          5                    4
 a. 15 ...
EXAMPLE 2
Use a calculator to estimate to the nearest hundredth.

                          5                    4
 a. 15 ...
EXAMPLE 2
Use a calculator to estimate to the nearest hundredth.

                          5                    4
 a. 15 ...
EXAMPLE 2
Use a calculator to estimate to the nearest hundredth.

                          5                    4
 a. 15 ...
EXAMPLE 2
Use a calculator to estimate to the nearest hundredth.

                          5                    4
 a. 15 ...
NOTICE



       1                   1                      1
            5                   4
15 = 15 ;       4829 = 482...
ROOT OF A POWER
       THEOREM

For all positive integers m > 1 and n ≥ 2, and all
          nonnegative real numbers x,
 ...
EXAMPLE 3
                 Simplify, for x ≥ 0.

                                             3
     4       8            ...
EXAMPLE 3
                     Simplify, for x ≥ 0.

                                                 3
     4       8    ...
EXAMPLE 3
                     Simplify, for x ≥ 0.

                                                 3
     4       8    ...
EXAMPLE 3
                     Simplify, for x ≥ 0.

                                                 3
     4       8    ...
EXAMPLE 3
                     Simplify, for x ≥ 0.

                                                 3
     4       8    ...
EXAMPLE 4
Suppose x ≥ 0. Rewrite using rational exponents.

           6                          34
      a.       x     ...
EXAMPLE 4
Suppose x ≥ 0. Rewrite using rational exponents.

           6                          34
      a.           x ...
EXAMPLE 4
Suppose x ≥ 0. Rewrite using rational exponents.

           6                          34
      a.           x ...
EXAMPLE 4
Suppose x ≥ 0. Rewrite using rational exponents.

           6                          34
      a.           x ...
EXAMPLE 4
Suppose x ≥ 0. Rewrite using rational exponents.

           6                          34
      a.           x ...
EXAMPLE 4
Suppose x ≥ 0. Rewrite using rational exponents.

           6                          34
      a.           x ...
EXAMPLE 5
   Rewrite using radical signs.

       1                            4
a. x                         b. x
       ...
EXAMPLE 5
      Rewrite using radical signs.

       1                               4
a. x                            b. ...
EXAMPLE 5
      Rewrite using radical signs.

       1                                 4
a. x                            b...
EXAMPLE 5
      Rewrite using radical signs.

       1                                  4
a. x                            ...
EXAMPLE 5
      Rewrite using radical signs.

       1                                  4
a. x                            ...
HOMEWORK



                     p. 497 #1-29




“The only way most people recognize their limits is by
         trespass...
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AA Section 8-4

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  • AA Section 8-4

    1. 1. SECTION 8-4 Radical Notation for nth Roots
    2. 2. WARM-UP What is the length of the side of a cube whose volume is 512 cm3?
    3. 3. WARM-UP What is the length of the side of a cube whose volume is 512 cm3?
    4. 4. WARM-UP What is the length of the side of a cube whose volume is 512 cm3? V = 512
    5. 5. WARM-UP What is the length of the side of a cube whose volume is 512 cm3? V = 512 3 V=s
    6. 6. WARM-UP What is the length of the side of a cube whose volume is 512 cm3? V = 512 3 V=s 3 s = 512
    7. 7. WARM-UP What is the length of the side of a cube whose volume is 512 cm3? V = 512 3 V=s 3 s = 512 1 1 3 (s ) = 5123 3
    8. 8. WARM-UP What is the length of the side of a cube whose volume is 512 cm3? V = 512 3 V=s 3 s = 512 1 1 3 (s ) = 5123 3 s = 8 cm
    9. 9. WARM-UP What is the length of the side of a cube whose volume is 512 cm3? V = 512 8 cm 3 V=s 8 cm 8 cm 3 s = 512 1 1 3 (s ) = 5123 3 s = 8 cm
    10. 10. n th ROOTS 1 x is the nth root of x n
    11. 11. n th ROOTS 1 x is the nth root of x n Square root of x:
    12. 12. n th ROOTS 1 x is the nth root of x n 1 x Square root of x: 2
    13. 13. n th ROOTS 1 x is the nth root of x n 1 x= x Square root of x: 2
    14. 14. n th ROOTS 1 x is the nth root of x n 1 x= x Square root of x: 2 Cube root of x:
    15. 15. n th ROOTS 1 x is the nth root of x n 1 x= x Square root of x: 2 1 x Cube root of x: 3
    16. 16. n th ROOTS 1 x is the nth root of x n 1 x= x Square root of x: 2 1 3 x= x Cube root of x: 3
    17. 17. n th ROOTS 1 x is the nth root of x n 1 x= x Square root of x: 2 1 3 x= x Cube root of x: 3 nth root of x:
    18. 18. n th ROOTS 1 x is the nth root of x n 1 x= x Square root of x: 2 1 3 x= x Cube root of x: 3 1 x nth root of x: n
    19. 19. n th ROOTS 1 x is the nth root of x n 1 x= x Square root of x: 2 1 3 x= x Cube root of x: 3 1 n x= x nth root of x: n
    20. 20. n th ROOTS 1 x is the nth root of x n 1 x= x Square root of x: 2 1 3 x= x Cube root of x: 3 1 n x= x nth root of x: n ***Notice: In radical notation, if there is no number in the “seat,” it is understood to be a square root
    21. 21. DEFINITION For any nonnegative real number x and any integer n ≥ 2, 1 n x=x n
    22. 22. DEFINITION For any nonnegative real number x and any integer n ≥ 2, 1 n x=x n (the nth root of x)
    23. 23. EXAMPLE 1 Evaluate without using a calculator. 3 6 a. 27 b. 64
    24. 24. EXAMPLE 1 Evaluate without using a calculator. 3 6 a. 27 b. 64 =3
    25. 25. EXAMPLE 1 Evaluate without using a calculator. 3 6 a. 27 b. 64 =3 =2
    26. 26. EXAMPLE 2 Use a calculator to estimate to the nearest hundredth. 5 4 a. 15 b. 4829 c. 2401
    27. 27. EXAMPLE 2 Use a calculator to estimate to the nearest hundredth. 5 4 a. 15 b. 4829 c. 2401 ≈ 3.87
    28. 28. EXAMPLE 2 Use a calculator to estimate to the nearest hundredth. 5 4 a. 15 b. 4829 c. 2401 ≈ 3.87
    29. 29. EXAMPLE 2 Use a calculator to estimate to the nearest hundredth. 5 4 a. 15 b. 4829 c. 2401 ≈ 3.87
    30. 30. EXAMPLE 2 Use a calculator to estimate to the nearest hundredth. 5 4 a. 15 b. 4829 c. 2401 ≈ 3.87 ≈ 5.45
    31. 31. EXAMPLE 2 Use a calculator to estimate to the nearest hundredth. 5 4 a. 15 b. 4829 c. 2401 ≈ 3.87 =7 ≈ 5.45
    32. 32. NOTICE 1 1 1 5 4 15 = 15 ; 4829 = 4829 ; 2401 = 2401 2 5 4
    33. 33. ROOT OF A POWER THEOREM For all positive integers m > 1 and n ≥ 2, and all nonnegative real numbers x, m ( x) m n m n x= =x n
    34. 34. EXAMPLE 3 Simplify, for x ≥ 0. 3 4 8 18 a. x b. x
    35. 35. EXAMPLE 3 Simplify, for x ≥ 0. 3 4 8 18 a. x b. x 1 8 = (x ) 4
    36. 36. EXAMPLE 3 Simplify, for x ≥ 0. 3 4 8 18 a. x b. x 1 8 = (x ) 4 2 =x
    37. 37. EXAMPLE 3 Simplify, for x ≥ 0. 3 4 8 18 a. x b. x 1 1 8 18 = (x ) = (x ) 3 4 2 =x
    38. 38. EXAMPLE 3 Simplify, for x ≥ 0. 3 4 8 18 a. x b. x 1 1 8 18 = (x ) = (x ) 3 4 2 6 =x =x
    39. 39. EXAMPLE 4 Suppose x ≥ 0. Rewrite using rational exponents. 6 34 a. x 2 b. x
    40. 40. EXAMPLE 4 Suppose x ≥ 0. Rewrite using rational exponents. 6 34 a. x 2 b. x 1 1 = (x ) 2 6
    41. 41. EXAMPLE 4 Suppose x ≥ 0. Rewrite using rational exponents. 6 34 a. x 2 b. x 1 1 = (x ) 2 6 1 =x 12
    42. 42. EXAMPLE 4 Suppose x ≥ 0. Rewrite using rational exponents. 6 34 a. x 2 b. x 1 1 1 1 2 = (x ) = ((x ) ) 3 2 6 4 1 =x 12
    43. 43. EXAMPLE 4 Suppose x ≥ 0. Rewrite using rational exponents. 6 34 a. x 2 b. x 1 1 1 1 2 = (x ) = ((x ) ) 3 2 6 4 1 1 1 = (x ) =x 3 2 12
    44. 44. EXAMPLE 4 Suppose x ≥ 0. Rewrite using rational exponents. 6 34 a. x 2 b. x 1 1 1 1 2 = (x ) = ((x ) ) 3 2 6 4 1 1 1 = (x ) =x 3 2 12 1 =x 6
    45. 45. EXAMPLE 5 Rewrite using radical signs. 1 4 a. x b. x 3 7
    46. 46. EXAMPLE 5 Rewrite using radical signs. 1 4 a. x b. x 3 7 3 =x
    47. 47. EXAMPLE 5 Rewrite using radical signs. 1 4 a. x b. x 3 7 3 7 4 =x =x
    48. 48. EXAMPLE 5 Rewrite using radical signs. 1 4 a. x b. x 3 7 3 7 4 =x =x or
    49. 49. EXAMPLE 5 Rewrite using radical signs. 1 4 a. x b. x 3 7 3 7 4 =x =x or 4 ( x) 7 =
    50. 50. HOMEWORK p. 497 #1-29 “The only way most people recognize their limits is by trespassing on them.” - Tom Morris

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