This document contains a section on radical notation for nth roots. It includes examples of evaluating nth roots with and without a calculator, simplifying expressions with nth roots using properties of radicals, and rewriting expressions between radical and rational exponent notation. The section concludes with examples rewriting expressions using only radical notation. The document provides explanations and step-by-step workings for each example. It aims to teach students how to manipulate and evaluate expressions involving 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