1) The document contains examples and theorems regarding nth roots. It discusses estimating the 12th root of 2, finding frequencies based on constant ratios, and the number of real roots. 2) One example shows that 1-i is a fourth root of -4 through expanding (1-i)^4. 3) Another example determines that the solution to x5=500 lies between the consecutive integers 3 and 4.

1. SECTION 7-6
nth Roots

2. WARM-UP
2 3 4
1. x =144 2. x = 8 3. x = 81
2 3
25 1
5. x = 6. x =
3
4. x = 64 64 27

3. WARM-UP
2 3 4
1. x =144 2. x = 8 3. x = 81
x = ±12
2 3
25 1
5. x = 6. x =
3
4. x = 64 64 27

4. WARM-UP
2 3 4
1. x =144 2. x = 8 3. x = 81
x=2
x = ±12
2 3
25 1
5. x = 6. x =
3
4. x = 64 64 27

5. WARM-UP
2 3 4
1. x =144 2. x = 8 3. x = 81
x=2
x = ±12 x = ±3
2 3
25 1
5. x = 6. x =
3
4. x = 64 64 27

6. WARM-UP
2 3 4
1. x =144 2. x = 8 3. x = 81
x=2
x = ±12 x = ±3
2 3
25 1
5. x = 6. x =
3
4. x = 64 64 27
x=4

7. WARM-UP
2 3 4
1. x =144 2. x = 8 3. x = 81
x=2
x = ±12 x = ±3
2 3
25 1
5. x = 6. x =
3
4. x = 64 64 27
5
x=±
x=4 8

8. WARM-UP
2 3 4
1. x =144 2. x = 8 3. x = 81
x=2
x = ±12 x = ±3
2 3
25 1
5. x = 6. x =
3
4. x = 64 64 27
5 1
x=± x=
x=4 8 3

9. th Roots
n
For integers n > 1, b is an nth root of x IFF bn = x

10. EXAMPLE 1
a. Estimate the 12th root of 2 (the constant ratio for a
piano)

11. EXAMPLE 1
a. Estimate the 12th root of 2 (the constant ratio for a
piano)
x12 = 2

12. EXAMPLE 1
a. Estimate the 12th root of 2 (the constant ratio for a
piano)
x12 = 2
12
y = x


y = 2


13. EXAMPLE 1

14. EXAMPLE 1

15. EXAMPLE 1

16. EXAMPLE 1

17. EXAMPLE 1

18. EXAMPLE 1

19. EXAMPLE 1

20. EXAMPLE 1
x ≈ ±1.0594631

21. EXAMPLE 1
b. Find the frequency of the F above the A with frequency
220 hertz.
http://www.smu.edu/totw/keybrd2.gif

22. EXAMPLE 1
b. Find the frequency of the F above the A with frequency
220 hertz.
http://www.smu.edu/totw/keybrd2.gif
The F is 8 steps above the A.

23. EXAMPLE 1

24. EXAMPLE 1
Fn = rFn−1, for int. n ≥1

25. EXAMPLE 1
Fn = rFn−1, for int. n ≥1
n−1
Fn = F0 r , for int. n ≥1

26. EXAMPLE 1
Fn = rFn−1, for int. n ≥1
n−1
Fn = F0 r , for int. n ≥1
8 −1
F8 = F0 (1.0594631)

27. EXAMPLE 1
Fn = rFn−1, for int. n ≥1
n−1
Fn = F0 r , for int. n ≥1
8 −1
F8 = F0 (1.0594631)
7
= 220(1.0594631)

28. EXAMPLE 1
Fn = rFn−1, for int. n ≥1
n−1
Fn = F0 r , for int. n ≥1
8 −1
F8 = F0 (1.0594631)
7
= 220(1.0594631)
≈ 329.6275692

29. EXAMPLE 1
Fn = rFn−1, for int. n ≥1
n−1
Fn = F0 r , for int. n ≥1
OR
8 −1
F8 = F0 (1.0594631)
7
= 220(1.0594631)
≈ 329.6275692

30. EXAMPLE 1
Fn = rFn−1, for int. n ≥1
1
n−1
Fn = F0 r , for int. n ≥1 8 −1
F8 = 220(2 )
12
OR
8 −1
F8 = F0 (1.0594631)
7
= 220(1.0594631)
≈ 329.6275692

31. EXAMPLE 1
Fn = rFn−1, for int. n ≥1
1
n−1
Fn = F0 r , for int. n ≥1 8 −1
F8 = 220(2 )
12
OR 7
8 −1
F8 = F0 (1.0594631) = 220(2 )
12
7
= 220(1.0594631)
≈ 329.6275692

32. EXAMPLE 1
Fn = rFn−1, for int. n ≥1
1
n−1
Fn = F0 r , for int. n ≥1 8 −1
F8 = 220(2 )
12
OR 7
8 −1
F8 = F0 (1.0594631) = 220(2 )
12
7
≈ 329.6275692
= 220(1.0594631)
≈ 329.6275692

33. EXAMPLE 1
Fn = rFn−1, for int. n ≥1
1
n−1
Fn = F0 r , for int. n ≥1 8 −1
F8 = 220(2 )
12
OR 7
8 −1
F8 = F0 (1.0594631) = 220(2 ) 12
7
≈ 329.6275692
= 220(1.0594631)
≈ 329.6275692
The frequency is about 330 hertz

34. 1/n Exponent Theorem
When x ≥ 0 and n is an integer greater than 1, x1/n is the
nth root of x

35. 1/n Exponent Theorem
When x ≥ 0 and n is an integer greater than 1, x1/n is the
nth root of x
1
x = square root
2

36. 1/n Exponent Theorem
When x ≥ 0 and n is an integer greater than 1, x1/n is the
nth root of x
1
x = square root
2
1
x = cube root
3

37. 1/n Exponent Theorem
When x ≥ 0 and n is an integer greater than 1, x1/n is the
nth root of x
1
x = square root
2
1
x = cube root
3
1
x = fourth root
4

38. EXAMPLE 2
Evaluate.
1
()
1 1 1
16
c. 4
a. 27 b. 25 d. 115
3 2 3
625

39. EXAMPLE 2
Evaluate.
1
()
1 1 1
16
c. 4
a. 27 b. 25 d. 115
3 2 3
625
3

40. EXAMPLE 2
Evaluate.
1
()
1 1 1
16
c. 4
a. 27 b. 25 d. 115
3 2 3
625
3 5

41. EXAMPLE 2
Evaluate.
1
()
1 1 1
16
c. 4
a. 27 b. 25 d. 115
3 2 3
625
2
3 5 5

42. EXAMPLE 2
Evaluate.
1
()
1 1 1
16
c. 4
a. 27 b. 25 d. 115
3 2 3
625
2
≈ 4.86
3 5 5

43. NUMBER OF REAL ROOTS
THEOREM

44. NUMBER OF REAL ROOTS
THEOREM
Every positive real number has two real nth roots
when n is even

45. NUMBER OF REAL ROOTS
THEOREM
Every positive real number has two real nth roots
when n is even
Every positive real number has one real nth root when
n is odd

46. NUMBER OF REAL ROOTS
THEOREM
Every positive real number has two real nth roots
when n is even
Every positive real number has one real nth root when
n is odd
Every negative real number has zero real nth roots
when n is even

47. NUMBER OF REAL ROOTS
THEOREM
Every positive real number has two real nth roots
when n is even
Every positive real number has one real nth root when
n is odd
Every negative real number has zero real nth roots
when n is even
Every negative real number has one real nth root
when n is odd

48. EXAMPLE 3
Show that 1- i is a fourth root of -4.

49. EXAMPLE 3
Show that 1- i is a fourth root of -4.
4
(1− i)

50. EXAMPLE 3
Show that 1- i is a fourth root of -4.
4
(1− i)
2 2
= (1− i) (1− i)

51. EXAMPLE 3
Show that 1- i is a fourth root of -4.
4
(1− i)
2 2
= (1− i) (1− i)
2 2
= (1− 2i + i )(1− 2i + i )

52. EXAMPLE 3
Show that 1- i is a fourth root of -4.
4
(1− i)
2 2
= (1− i) (1− i)
2 2
= (1− 2i + i )(1− 2i + i )
= (1− 2i −1)(1− 2i −1)

53. EXAMPLE 3
Show that 1- i is a fourth root of -4.
4
(1− i)
2 2
= (1− i) (1− i)
2 2
= (1− 2i + i )(1− 2i + i )
= (1− 2i −1)(1− 2i −1)
= (−2i)(−2i)

54. EXAMPLE 3
Show that 1- i is a fourth root of -4.
4
(1− i)
2 2
= (1− i) (1− i)
2 2
= (1− 2i + i )(1− 2i + i )
= (1− 2i −1)(1− 2i −1)
= (−2i)(−2i)
2
= 4i

55. EXAMPLE 3
Show that 1- i is a fourth root of -4.
4
(1− i)
2 2
= (1− i) (1− i)
2 2
= (1− 2i + i )(1− 2i + i )
= (1− 2i −1)(1− 2i −1)
= (−2i)(−2i)
2
= 4i
= −4

56. EXAMPLE 4
Between which two consecutive integers is the real
solution to x5 = 500? Do not use a calculator.

57. EXAMPLE 4
Between which two consecutive integers is the real
solution to x5 = 500? Do not use a calculator.
35

58. EXAMPLE 4
Between which two consecutive integers is the real
solution to x5 = 500? Do not use a calculator.
35 = (3)(3)(3)(3)(3)

59. EXAMPLE 4
Between which two consecutive integers is the real
solution to x5 = 500? Do not use a calculator.
35 = (3)(3)(3)(3)(3) = 243

60. EXAMPLE 4
Between which two consecutive integers is the real
solution to x5 = 500? Do not use a calculator.
35 = (3)(3)(3)(3)(3) = 243
45

61. EXAMPLE 4
Between which two consecutive integers is the real
solution to x5 = 500? Do not use a calculator.
35 = (3)(3)(3)(3)(3) = 243
45 = (4)(4)(4)(4)(4)

62. EXAMPLE 4
Between which two consecutive integers is the real
solution to x5 = 500? Do not use a calculator.
35 = (3)(3)(3)(3)(3) = 243
45 = (4)(4)(4)(4)(4) = 1024

63. EXAMPLE 4
Between which two consecutive integers is the real
solution to x5 = 500? Do not use a calculator.
35 = (3)(3)(3)(3)(3) = 243
45 = (4)(4)(4)(4)(4) = 1024
The fifth root of 500 is between 3 and 4.

64. HOMEWORK

65. HOMEWORK
p. 456 #1-28

66. HOMEWORK
p. 456 #1-28
“If I have ever made any valuable discoveries, it has been
owing more to patient attention, than to any other
talent.” - Isaac Newton