The document provides examples and explanations of how to simplify radicals by breaking them into perfect powers that match the given root. It also defines the mean and geometric mean. The mean is calculated by adding all the values and dividing by the total number of values. The geometric mean is calculated by multiplying all the values and taking the nth root, where n is the total number of values. An example is provided to find the geometric mean of the integers from 1 to 10.

1. Section 8-5
Products with Radicals

2. Warm-up
1. A rectangle has an area of 10 ft2. Its length is 20 ft.
Find the width.
5
2. (3t ) = ?

3. Warm-up
1. A rectangle has an area of 10 ft2. Its length is 20 ft.
Find the width.
A = 10 ft2 w =?
l = 20 ft
5
2. (3t ) = ?

4. Warm-up
1. A rectangle has an area of 10 ft2. Its length is 20 ft.
Find the width.
A = 10 ft2 w =?
A = lw
l = 20 ft
5
2. (3t ) = ?

5. Warm-up
1. A rectangle has an area of 10 ft2. Its length is 20 ft.
Find the width.
A = 10 ft2 w =?
A = lw
l = 20 ft
10 = w 20
5
2. (3t ) = ?

6. Warm-up
1. A rectangle has an area of 10 ft2. Its length is 20 ft.
Find the width.
A = 10 ft2 w =?
A = lw
l = 20 ft
10 = w 20
10
w= 20
5
2. (3t ) = ?

7. Warm-up
1. A rectangle has an area of 10 ft2. Its length is 20 ft.
Find the width.
A = 10 ft2 w =?
A = lw
l = 20 ft
10 = w 20
10
w= 20
•
20 20
5
2. (3t ) = ?

8. Warm-up
1. A rectangle has an area of 10 ft2. Its length is 20 ft.
Find the width.
A = 10 ft2 w =?
A = lw
l = 20 ft
10 = w 20
10
w= 20 10 20
• =
20 20
20
5
2. (3t ) = ?

9. Warm-up
1. A rectangle has an area of 10 ft2. Its length is 20 ft.
Find the width.
A = 10 ft2 w =?
A = lw
l = 20 ft
10 = w 20
10
w= 20 10 20 20
• = =
20 20 2
20
5
2. (3t ) = ?

10. Warm-up
1. A rectangle has an area of 10 ft2. Its length is 20 ft.
Find the width.
A = 10 ft2 w =?
A = lw
l = 20 ft
10 = w 20
10
w= 20 10 20 20 4•5
• = = =
20 20 2 2
20
5
2. (3t ) = ?

11. Warm-up
1. A rectangle has an area of 10 ft2. Its length is 20 ft.
Find the width.
A = 10 ft2 w =?
A = lw
l = 20 ft
10 = w 20
10
w= 20 10 20 20 4•5 25
• = = = =
20 20 2 2 2
20
5
2. (3t ) = ?

12. Warm-up
1. A rectangle has an area of 10 ft2. Its length is 20 ft.
Find the width.
A = 10 ft2 w =?
A = lw
l = 20 ft
10 = w 20
10
w= 20 10 20 20 4•5 25
• = = = = =5
20 20 2 2 2
20
5
2. (3t ) = ?

13. Warm-up
1. A rectangle has an area of 10 ft2. Its length is 20 ft.
Find the width.
A = 10 ft2 w =?
A = lw
l = 20 ft
10 = w 20
10
w= 20 10 20 20 4•5 25
• = = = = = 5 ft
20 20 2 2 2
20
5
2. (3t ) = ?

14. Warm-up
1. A rectangle has an area of 10 ft2. Its length is 20 ft.
Find the width.
A = 10 ft2 w =?
A = lw
l = 20 ft
10 = w 20
10
w= 20 10 20 20 4•5 25
• = = = = = 5 ft
20 20 2 2 2
20
5
2. (3t ) = ?
5 5 5
(3t ) = 3 t

15. Warm-up
1. A rectangle has an area of 10 ft2. Its length is 20 ft.
Find the width.
A = 10 ft2 w =?
A = lw
l = 20 ft
10 = w 20
10
w= 20 10 20 20 4•5 25
• = = = = = 5 ft
20 20 2 2 2
20
5
2. (3t ) = ?
5 5 5 5
(3t ) = 3 t = 243t

16. How do you simplify radicals?

17. m m m
Recall: (xy ) = x y

18. m m m
Recall: (xy ) = x y
1 1 1
So: (xy ) = x y
n n n

19. m m m
Recall: (xy ) = x y
1 1 1
So: (xy ) = x y
n n n
Thus: n
n
xy = x n y

20. m m m
Recall: (xy ) = x y
1 1 1
So: (xy ) = x y
n n n
Thus: n
n
xy = x n y
Now we can break up and simplify a radical by factoring the
number inside!

21. Root of a Product
Theorem
For all nonnegative real numbers x and y, and for any
integer n ≥ 2,
1 1 1
n
n n
(xy ) = x y or xy = x y
n n n

22. Example 1
Find the product.
3
3
4 • 16

23. Example 1
Find the product.
3
3
4 • 16
3
= 4 •16

24. Example 1
Find the product.
3
3
4 • 16
3
= 4 •16
3
= 64

25. Example 1
Find the product.
3
3
4 • 16
3
= 4 •16
3
= 64
=4

26. Example 2
Assume t > 0. Write as a single radical.
7 7
4 2
2t • 7t

27. Example 2
Assume t > 0. Write as a single radical.
7 7
4 2
2t • 7t
7 4 2
= 2t • 7t

28. Example 2
Assume t > 0. Write as a single radical.
7 7
4 2
2t • 7t
7 4 2
= 2t • 7t
7 6
= 14t

29. Simplifying Radicals
Break up the radical into perfect powers that
match the given root

30. Example 3
Simplify
3 635
128a b c

31. Example 3
Simplify
3 635
128a b c
3 6332
= 64 • 2a b c c

32. Example 3
Simplify
3 635
128a b c
3 6332
= 64 • 2a b c c
3
3 633 2
= 64a b c • 2c

33. Example 3
Simplify
3 635
128a b c
3 6332
= 64 • 2a b c c
3
3 633 2
= 64a b c • 2c
3
2 2
= 4a bc 2c

34. Simplifying an n th Root
When you factor the expression in the radical into
perfect nth powers

35. Example 4
Simplify
83
4
81x y

36. Example 4
Simplify
83
4
81x y
4 8 3
4
= 81x • y

37. Example 4
Simplify
83
4
81x y
4 8 3
4
= 81x • y
24 3
y
= 3x

38. Example 5
Simplify
57
3
54x y

39. Example 5
Simplify
57
3
54x y
3 26
3
= 27 • 2x x y y

40. Example 5
Simplify
57
3
54x y
3 26
3
= 27 • 2x x y y
36 2
3 3
= 27x y • 2x y

41. Example 5
Simplify
57
3
54x y
3 26
3
= 27 • 2x x y y
36 2
3 3
= 27x y • 2x y
23 2
2x y
= 3xy

42. Mean:

43. Mean: Also known as the average

44. Mean: Also known as the average
Add up all the values, then divide by total number of
values

45. Mean: Also known as the average
Add up all the values, then divide by total number of
values
Arithmetic mean

46. Mean: Also known as the average
Add up all the values, then divide by total number of
values
Arithmetic mean (because we’re adding)

47. Mean: Also known as the average
Add up all the values, then divide by total number of
values
Arithmetic mean (because we’re adding)
Geometric Mean:

48. Mean: Also known as the average
Add up all the values, then divide by total number of
values
Arithmetic mean (because we’re adding)
Geometric Mean: Multiply all the values in a set of numbers
then take the nth root, where n is the total number
of values in the set

49. Example 6
Find the geometric mean of the integers 1 to 10 to the
nearest hundredth

50. Example 6
Find the geometric mean of the integers 1 to 10 to the
nearest hundredth
1• 2 • 3 • 4 • 5 • 6 • 7 • 8• 9 •10

51. Example 6
Find the geometric mean of the integers 1 to 10 to the
nearest hundredth
1• 2 • 3 • 4 • 5 • 6 • 7 • 8• 9 •10
= 3628800

52. Example 6
Find the geometric mean of the integers 1 to 10 to the
nearest hundredth
1• 2 • 3 • 4 • 5 • 6 • 7 • 8• 9 •10
= 3628800
10
3628800

53. Example 6
Find the geometric mean of the integers 1 to 10 to the
nearest hundredth
1• 2 • 3 • 4 • 5 • 6 • 7 • 8• 9 •10
= 3628800
10
3628800
≈ 4.53

54. Homework
p. 503 #1 - 29
“While one person hesitates because he feels inferior, the
other is busy making mistakes and becoming superior.” -
Henry C. Link