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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