Radical Notation for nth Roots

1. SECTION 8-4
Radical Notation for nth Roots

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

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

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

5. WARM-UP
What is the length of the side of a cube whose volume
is 512 cm3?
V = 512
3
V=s

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

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

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

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

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

22. DEFINITION
For any nonnegative real number x and any integer n ≥ 2,
1
n
x=x n
(the nth root of x)

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

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

25. EXAMPLE 1
Evaluate without using a calculator.
3 6
a. 27 b. 64
=3 =2

26. EXAMPLE 2
Use a calculator to estimate to the nearest hundredth.
5 4
a. 15 b. 4829 c. 2401

27. EXAMPLE 2
Use a calculator to estimate to the nearest hundredth.
5 4
a. 15 b. 4829 c. 2401
≈ 3.87

28. EXAMPLE 2
Use a calculator to estimate to the nearest hundredth.
5 4
a. 15 b. 4829 c. 2401
≈ 3.87

29. EXAMPLE 2
Use a calculator to estimate to the nearest hundredth.
5 4
a. 15 b. 4829 c. 2401
≈ 3.87

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. 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. NOTICE
1 1 1
5 4
15 = 15 ; 4829 = 4829 ; 2401 = 2401
2 5 4

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. EXAMPLE 3
Simplify, for x ≥ 0.
3
4 8 18
a. x b. x

35. EXAMPLE 3
Simplify, for x ≥ 0.
3
4 8 18
a. x b. x
1
8
= (x ) 4

36. EXAMPLE 3
Simplify, for x ≥ 0.
3
4 8 18
a. x b. x
1
8
= (x ) 4
2
=x

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. 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. EXAMPLE 4
Suppose x ≥ 0. Rewrite using rational exponents.
6 34
a. x 2
b. x

40. EXAMPLE 4
Suppose x ≥ 0. Rewrite using rational exponents.
6 34
a. x 2
b. x
1 1
= (x ) 2 6

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. 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. 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. 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. EXAMPLE 5
Rewrite using radical signs.
1 4
a. x b. x
3 7

46. EXAMPLE 5
Rewrite using radical signs.
1 4
a. x b. x
3 7
3
=x

47. EXAMPLE 5
Rewrite using radical signs.
1 4
a. x b. x
3 7
3 7 4
=x =x

48. EXAMPLE 5
Rewrite using radical signs.
1 4
a. x b. x
3 7
3 7 4
=x =x
or

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. HOMEWORK
p. 497 #1-29
“The only way most people recognize their limits is by
trespassing on them.” - Tom Morris