Quotients with Radicals

1. SECTION 8-6
Quotients with Radicals

2. WARM-UP
Simplify.
1 2
a. b.
1
3 .6

3. WARM-UP
Simplify.
1 2
a. b.
1
3 .6
1 3
= •
1
3 3

4. WARM-UP
Simplify.
1 2
a. b.
1
3 .6
1 3
= •
1
3 3
=3

5. WARM-UP
Simplify.
1 2
a. b.
1
3 .6
1 3 2 10
= • = •
.6 10
1
3 3
=3

6. WARM-UP
Simplify.
1 2
a. b.
1
3 .6
1 3 2 10
= • = •
.6 10
1
3 3
20
=3 =
6

7. WARM-UP
Simplify.
1 2
a. b.
1
3 .6
1 3 2 10
= • = •
.6 10
1
3 3
20 10
=3 = =
6 3

8. RATIONALIZE THE
DENOMINATOR
Rewriting a fraction without an irrational number in
the denominator

9. EXAMPLE 1
Rationalize, then simplify.
30
10

10. EXAMPLE 1
Rationalize, then simplify.
30
10
30 10
= •
10 10

11. EXAMPLE 1
Rationalize, then simplify.
30
10
30 10
= •
10 10
30 10
=
10

12. EXAMPLE 1
Rationalize, then simplify.
30
10
30 10
= •
10 10
30 10
=
10
= 3 10

13. PROCESS
To get rid of the radical in the denominator, multiply the
numerator and denominator by the radical n times,
where n is the root of the radical

14. EXAMPLE 2
Simplify.
1 20
a. b.
7
5b 25n

15. EXAMPLE 2
Simplify.
1 20
a. b.
7
5b 25n
1
=
5b

16. EXAMPLE 2
Simplify.
1 20
a. b.
7
5b 25n
1
=
5b
1 5b
= •
5b 5b

17. EXAMPLE 2
Simplify.
1 20
a. b.
7
5b 25n
1
=
5b
1 5b
= •
5b 5b
5b
=
5b

18. EXAMPLE 2
Simplify.
1 20
a. b.
7
5b 25n
1 20 25n7
= = •
5b 25n 7
25n 7
1 5b
= •
5b 5b
5b
=
5b

19. EXAMPLE 2
Simplify.
1 20
a. b.
7
5b 25n
1 20 25n 7
20 25n 7
= = • =
5b 25n 7
25n 7
25n 7
1 5b
= •
5b 5b
5b
=
5b

20. EXAMPLE 2
Simplify.
1 20
a. b.
7
5b 25n
1 20 25n 7
20 25n 7
= = • =
5b 25n 7
25n 7
25n 7
1 5b 20 25n 6
n
= • =
5b 5b 25n 7
5b
=
5b

21. EXAMPLE 2
Simplify.
1 20
a. b.
7
5b 25n
1 20 25n 7
20 25n 7
= = • =
5b 25n 7
25n 7
25n 7
3
1 5b 20 25n 6
n 20 • 5n n
= • = = 7
5b 5b 25n 7
25n
5b
=
5b

22. EXAMPLE 2
Simplify.
1 20
a. b.
7
5b 25n
1 20 25n 7
20 25n 7
= = • =
5b 25n 7
25n 7
25n 7
3
1 5b 20 25n 6
n 20 • 5n n
= • = = 7
5b 5b 25n 7
25n
3
5b 100n n
= = 7
5b 25n

23. EXAMPLE 2
Simplify.
1 20
a. b.
7
5b 25n
1 20 25n 7
20 25n 7
= = • =
5b 25n 7
25n 7
25n 7
3
1 5b 20 25n 6
n 20 • 5n n
= • = = 7
5b 5b 25n 7
25n
3
5b 100n n 4 n
= = 7
= 4
5b 25n n

24. RECALL
The complex conjugate of a complex number a + bi is
a - bi.

25. RECALL
The complex conjugate of a complex number a + bi is
a - bi.
3 + 2i
Simplify
6 − 4i

26. RECALL
The complex conjugate of a complex number a + bi is
a - bi.
3 + 2i
Simplify
6 − 4i
3 + 2i 6 + 4i
= •
6 − 4i 6 + 4i

27. RECALL
The complex conjugate of a complex number a + bi is
a - bi.
3 + 2i
Simplify
6 − 4i
3 + 2i 6 + 4i 18 +12i +12i + 8i 2
= • =
6 − 4i 6 + 4i 36 + 24i − 24i −16i 2

28. RECALL
The complex conjugate of a complex number a + bi is
a - bi.
3 + 2i
Simplify
6 − 4i
3 + 2i 6 + 4i 18 +12i +12i + 8i 2
18 + 24i − 8
= • = =
6 − 4i 6 + 4i 36 + 24i − 24i −16i 2
36 +16

29. RECALL
The complex conjugate of a complex number a + bi is
a - bi.
3 + 2i
Simplify
6 − 4i
3 + 2i 6 + 4i 18 +12i +12i + 8i 2
18 + 24i − 8
= • = =
6 − 4i 6 + 4i 36 + 24i − 24i −16i 2
36 +16
10 + 24i
=
52

30. RECALL
The complex conjugate of a complex number a + bi is
a - bi.
3 + 2i
Simplify
6 − 4i
3 + 2i 6 + 4i 18 +12i +12i + 8i 2
18 + 24i − 8
= • = =
6 − 4i 6 + 4i 36 + 24i − 24i −16i 2
36 +16
10 + 24i 5 6
= = + i
52 26 13

31. EXAMPLE 3
Simplify.
2+3 5
2−3 5

32. EXAMPLE 3
Simplify.
2+3 5
2−3 5
2+3 5 2+3 5
= •
2−3 5 2+3 5

33. EXAMPLE 3
Simplify.
2+3 5
2−3 5
2+3 5 2+3 5 4 + 6 5 + 6 5 + 9(5)
= • =
2−3 5 2+3 5 4 + 6 5 − 6 5 − 9(5)

34. EXAMPLE 3
Simplify.
2+3 5
2−3 5
2+3 5 2+3 5 4 + 6 5 + 6 5 + 9(5) 4 +12 5 + 45
= • = =
2 − 3 5 2 + 3 5 4 + 6 5 − 6 5 − 9(5) 4 − 45

35. EXAMPLE 3
Simplify.
2+3 5
2−3 5
2+3 5 2+3 5 4 + 6 5 + 6 5 + 9(5) 4 +12 5 + 45
= • = =
2 − 3 5 2 + 3 5 4 + 6 5 − 6 5 − 9(5) 4 − 45
49 +12 5
=
−41

36. HOMEWORK
p. 508 #1-27
“The reason most people never reach their goals is that
they don’t believe them, or ever seriously consider
them as believeable or achievable. Winners can tell you
where they are going, what they plan to do along the
way, and who will be sharing the adventure with them.”
- Denis Watley