1. More on Algebra of Radicals
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2. Remember that x·y = x·y, x·x = x
More on Algebra of Radicals
y
x
y
x
 =,

3. Remember that x·y = x·y,
3*3 *2* 2 *2 *3 *2
x·x = x
More on Algebra of Radicals
Example A. Simplify
a.
y
x
y
x
 =,

4. Remember that x·y = x·y,
3*3 *2* 2 *2 *3 *2
x·x = x
More on Algebra of Radicals
Example A. Simplify
a.
y
x
y
x
 =,

5. Remember that x·y = x·y,
3*3 *2* 2 *2 *3 *2
= 3 * 2 *3 *3 * 2 *2 *2
x·x = x
More on Algebra of Radicals
Example A. Simplify
a.
y
x
y
x
 =,

6. Remember that x·y = x·y,
3*3 *2* 2 *2 *3 *2
= 3 * 2 *3 *3 * 2 *2 *2
x·x = x
3
More on Algebra of Radicals
Example A. Simplify
2
a.
y
x
y
x
 =,

7. Remember that x·y = x·y,
3*3 *2* 2 *2 *3 *2
= 3 * 2 *3 *3 * 2 *2 *2
= 3 * 3 * 2 * 2 * 2
x·x = x
3
More on Algebra of Radicals
Example A. Simplify
2
a.
y
x
y
x
 =,

8. Remember that x·y = x·y,
3*3 *2* 2 *2 *3 *2
= 3 * 2 *3 *3 * 2 *2 *2
= 3 * 3 * 2 * 2 * 2
= 362
x·x = x
3
More on Algebra of Radicals
Example A. Simplify
2
a.
y
x
y
x
 =,

9. Remember that x·y = x·y,
3*3 *2* 2 *2 *3 *2
= 3 * 2 *3 *3 * 2 *2 *2
= 3 * 3 * 2 * 2 * 2
= 362
x·x = x
3
More on Algebra of Radicals
Example A. Simplify
2
b. 12 (3 + 32)
a.
y
x
y
x
 =,

10. Remember that x·y = x·y,
3*3 *2* 2 *2 *3 *2
= 3 * 2 *3 *3 * 2 *2 *2
= 3 * 3 * 2 * 2 * 2
= 362
x·x = x
3
More on Algebra of Radicals
Example A. Simplify
2
b. 12 (3 + 32)
= 123 + 3122
a.
y
x
y
x
 =,

11. Remember that x·y = x·y,
3*3 *2* 2 *2 *3 *2
= 3 * 2 *3 *3 * 2 *2 *2
= 3 * 3 * 2 * 2 * 2
= 362
x·x = x
3
More on Algebra of Radicals
Example A. Simplify
2
b. 12 (3 + 32)
= 123 + 3122
= 36+ 324
a.
y
x
y
x
 =,

12. Remember that x·y = x·y,
3*3 *2* 2 *2 *3 *2
= 3 * 2 *3 *3 * 2 *2 *2
= 3 * 3 * 2 * 2 * 2
= 362
x·x = x
3
More on Algebra of Radicals
Example A. Simplify
2
b. 12 (3 + 32)
= 123 + 3122
= 36+ 324
= 6 + 3√4*6
a.
y
x
y
x
 =,

13. Remember that x·y = x·y,
3*3 *2* 2 *2 *3 *2
= 3 * 2 *3 *3 * 2 *2 *2
= 3 * 3 * 2 * 2 * 2
= 362
x·x = x
3
More on Algebra of Radicals
Example A. Simplify
2
b. 12 (3 + 32)
= 123 + 3122
= 36+ 324
= 6 + 3√4*6
= 6 + 3*26
a.
y
x
y
x
 =,

14. Remember that x·y = x·y,
3*3 *2* 2 *2 *3 *2
= 3 * 2 *3 *3 * 2 *2 *2
= 3 * 3 * 2 * 2 * 2
= 362
x·x = x
3
More on Algebra of Radicals
Example A. Simplify
2
b. 12 (3 + 32)
= 123 + 3122
= 36+ 324
= 6 + 3√4*6
= 6 + 3*26
= 6 + 66
a.
y
x
y
x
 =,

15. Remember that x·y = x·y,
3*3 *2* 2 *2 *3 *2
= 3 * 2 *3 *3 * 2 *2 *2
= 3 * 3 * 2 * 2 * 2
= 362
x·x = x
3
More on Algebra of Radicals
Example A. Simplify
2
b. 12 (3 + 32)
= 123 + 3122
= 36+ 324
= 6 + 3√4*6
= 6 + 3*26
= 6 + 66
a.
(Remember 6 + 6√6 = 12√6 because they are not
like-terms.)
y
x
y
x
 =,

16. c. (33 – 22)(23 + 32)
More on Algebra of Radicals

17. c. (33 – 22)(23 + 32)
= 33*23
More on Algebra of Radicals

18. c. (33 – 22)(23 + 32)
= 33*23 + 33*32
More on Algebra of Radicals

19. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23
More on Algebra of Radicals

20. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals

21. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
3

22. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
3 √6 √6

23. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
3 2√6 √6

24. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
= 18 + 96 – 46 – 12
3 2√6 √6

25. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
= 18 + 96 – 46 – 12
= 6 + 56
3 2√6 √6

26. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
= 18 + 96 – 46 – 12
= 6 + 56
3 2√6 √6
Conjugates

27. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
= 18 + 96 – 46 – 12
= 6 + 56
3 2√6 √6
Conjugates
We call x + y, x – y the conjugate of each other.

28. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
= 18 + 96 – 46 – 12
= 6 + 56
3 2√6 √6
Conjugates
We call x + y, x – y the conjugate of each other.
For example, the conjugate of 3 – 25 is 3 + 25,

29. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
= 18 + 96 – 46 – 12
= 6 + 56
3 2√6 √6
Conjugates
We call x + y, x – y the conjugate of each other.
For example, the conjugate of 3 – 25 is 3 + 25,
the conjugate of 5 + 22 is 5 – 22.

30. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
= 18 + 96 – 46 – 12
= 6 + 56
3 2√6 √6
Conjugates
We call x + y, x – y the conjugate of each other.
For example, the conjugate of 3 – 25 is 3 + 25,
the conjugate of 5 + 22 is 5 – 22.
The importance of conjugate pair is that
(x + y)(x – y) = x2 – y2

31. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
= 18 + 96 – 46 – 12
= 6 + 56
3 2√6 √6
Conjugates
We call x + y, x – y the conjugate of each other.
For example, the conjugate of 3 – 25 is 3 + 25,
the conjugate of 5 + 22 is 5 – 22.
The importance of conjugate pair is that
(x + y)(x – y) = x2 – y2
Example B. Multiply the following conjugates.
a. (3 – 25)(3 + 25)

32. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
= 18 + 96 – 46 – 12
= 6 + 56
3 2√6 √6
Conjugates
We call x + y, x – y the conjugate of each other.
For example, the conjugate of 3 – 25 is 3 + 25,
the conjugate of 5 + 22 is 5 – 22.
The importance of conjugate pair is that
(x + y)(x – y) = x2 – y2
Example B. Multiply the following conjugates.
a. (3 – 25)(3 + 25)
= 32 – (25)2

33. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
= 18 + 96 – 46 – 12
= 6 + 56
3 2√6 √6
Conjugates
We call x + y, x – y the conjugate of each other.
For example, the conjugate of 3 – 25 is 3 + 25,
the conjugate of 5 + 22 is 5 – 22.
The importance of conjugate pair is that
(x + y)(x – y) = x2 – y2
Example B. Multiply the following conjugates.
a. (3 – 25)(3 + 25)
= 32 – (25)2
= 9 – 4*5

34. b.(5 + 22)(3 – 7)(5 – 22)(3 + 7)
More on Algebra of Radicals

35. b.(5 + 22)(3 – 7)(5 – 22)(3 + 7)
=(5 + 22)(5 – 22)(3 – 7)(3 + 7)
More on Algebra of Radicals

36. b.(5 + 22)(3 – 7)(5 – 22)(3 + 7)
=(5 + 22)(5 – 22)(3 – 7)(3 + 7)
= ((5)2 – (22)2) (32 – (7)2)
More on Algebra of Radicals

37. b.(5 + 22)(3 – 7)(5 – 22)(3 + 7)
=(5 + 22)(5 – 22)(3 – 7)(3 + 7)
= ((5)2 – (22)2) (32 – (7)2)
= (5 – 8)(9 – 7)
More on Algebra of Radicals

38. b.(5 + 22)(3 – 7)(5 – 22)(3 + 7)
=(5 + 22)(5 – 22)(3 – 7)(3 + 7)
= ((5)2 – (22)2) (32 – (7)2)
= (5 – 8)(9 – 7)
= –3*2
= –6
More on Algebra of Radicals

39. b.(5 + 22)(3 – 7)(5 – 22)(3 + 7)
=(5 + 22)(5 – 22)(3 – 7)(3 + 7)
= ((5)2 – (22)2) (32 – (7)2)
= (5 – 8)(9 – 7)
= –3*2
= –6
More on Algebra of Radicals
Conjugates are used to rationalize the denominator,
i.e. to rewrite a fraction with square-root term(s) in the
denominator so it does not contain any radical.

40. b.(5 + 22)(3 – 7)(5 – 22)(3 + 7)
=(5 + 22)(5 – 22)(3 – 7)(3 + 7)
= ((5)2 – (22)2) (32 – (7)2)
= (5 – 8)(9 – 7)
= –3*2
= –6
More on Algebra of Radicals
Conjugates are used to rationalize the denominator,
i.e. to rewrite a fraction with square-root term(s) in the
denominator so it does not contain any radical.
Example C. Rationalize the following expressions.
2
2 – 35
a.

41. b.(5 + 22)(3 – 7)(5 – 22)(3 + 7)
=(5 + 22)(5 – 22)(3 – 7)(3 + 7)
= ((5)2 – (22)2) (32 – (7)2)
= (5 – 8)(9 – 7)
= –3*2
= –6
More on Algebra of Radicals
Conjugates are used to rationalize the denominator,
i.e. to rewrite a fraction with square-root term(s) in the
denominator so it does not contain any radical.
Example C. Rationalize the following expressions.
2
2 – 35
a.
Multiply the top and bottom by the conjugate
of the denominator.

42. b.(5 + 22)(3 – 7)(5 – 22)(3 + 7)
=(5 + 22)(5 – 22)(3 – 7)(3 + 7)
= ((5)2 – (22)2) (32 – (7)2)
= (5 – 8)(9 – 7)
= –3*2
= –6
More on Algebra of Radicals
Conjugates are used to rationalize the denominator,
i.e. to rewrite a fraction with square-root term(s) in the
denominator so it does not contain any radical.
Example C. Rationalize the following expressions.
2
2 – 35
a.
=
2
(2 – 35)
Multiply the top and bottom by the conjugate
of the denominator.

43. b.(5 + 22)(3 – 7)(5 – 22)(3 + 7)
=(5 + 22)(5 – 22)(3 – 7)(3 + 7)
= ((5)2 – (22)2) (32 – (7)2)
= (5 – 8)(9 – 7)
= –3*2
= –6
More on Algebra of Radicals
Conjugates are used to rationalize the denominator,
i.e. to rewrite a fraction with square-root term(s) in the
denominator so it does not contain any radical.
Example C. Rationalize the following expressions.
2
2 – 35
a.
=
2
(2 – 35) (2 + 35)
(2 + 35)
Multiply the top and bottom by the conjugate
of the denominator.

44. b.(5 + 22)(3 – 7)(5 – 22)(3 + 7)
=(5 + 22)(5 – 22)(3 – 7)(3 + 7)
= ((5)2 – (22)2) (32 – (7)2)
= (5 – 8)(9 – 7)
= –3*2
= –6
More on Algebra of Radicals
Conjugates are used to rationalize the denominator,
i.e. to rewrite a fraction with square-root term(s) in the
denominator so it does not contain any radical.
Example C. Rationalize the following expressions.
2
2 – 35
a.
=
2
(2 – 35) (2 + 35)
(2 + 35)
Multiply the top and bottom by the conjugate
of the denominator.
(2)2 – (35)2 = 4 – 45 = –41

45. b.(5 + 22)(3 – 7)(5 – 22)(3 + 7)
=(5 + 22)(5 – 22)(3 – 7)(3 + 7)
= ((5)2 – (22)2) (32 – (7)2)
= (5 – 8)(9 – 7)
= –3*2
= –6
More on Algebra of Radicals
Conjugates are used to rationalize the denominator,
i.e. to rewrite a fraction with square-root term(s) in the
denominator so it does not contain any radical.
Example C. Rationalize the following expressions.
2
2 – 35
a.
=
2
(2 – 35) (2 + 35)
(2 + 35)
Multiply the top and bottom by the conjugate
of the denominator.
= 4 + 6√5
– 41
(2)2 – (35)2 = 4 – 45 = –41

46. b.
5 – 23
3 + 43
More on Algebra of Radicals

47. b.
5 – 23
3 + 43
More on Algebra of Radicals
Multiply the top and bottom by the conjugate
of the denominator.

48. b.
5 – 23
3 + 43
=
(3 – 43)
(3 – 43)
·
More on Algebra of Radicals
(5 – 23)
(3 + 43)
Multiply the top and bottom by the conjugate
of the denominator.

49. b.
5 – 23
3 + 43
=
(3 – 43)
(3 – 43)
·
More on Algebra of Radicals
(5 – 23)
(3 + 43)
Multiply the top and bottom by the conjugate
of the denominator.
(3)2 – (43)2

50. b.
5 – 23
3 + 43
=
(3 – 43)
(3 – 43)
·
More on Algebra of Radicals
(5 – 23)
(3 + 43)
Multiply the top and bottom by the conjugate
of the denominator.
(3)2 – (43)2 = 9 – 48 = –39

51. b.
5 – 23
3 + 43
=
(3 – 43)
(3 – 43)
·
(5 – 23) (3 – 43)
More on Algebra of Radicals
(5 – 23)
(3 + 43)
Multiply the top and bottom by the conjugate
of the denominator.
(3)2 – (43)2 = 9 – 48 = –39
= –39

52. b.
5 – 23
3 + 43
=
(3 – 43)
(3 – 43)
·
(5 – 23) (3 – 43)
= –39
More on Algebra of Radicals
(5 – 23)
(3 + 43)
Multiply the top and bottom by the conjugate
of the denominator.
(3)2 – (43)2 = 9 – 48 = –39
= –39
15 – 63 – 203 + 24

53. b.
5 – 23
3 + 43
=
(3 – 43)
(3 – 43)
·
(5 – 23) (3 – 43)
= –39
=
39 – 263
–39
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(5 – 23)
(3 + 43)
Multiply the top and bottom by the conjugate
of the denominator.
(3)2 – (43)2 = 9 – 48 = –39
= –39
15 – 63 – 203 + 24

54. b.
5 – 23
3 + 43
=
(3 – 43)
(3 – 43)
·
(5 – 23) (3 – 43)
= –39
=
39 – 263
–39
More on Algebra of Radicals
(5 – 23)
(3 + 43)
Multiply the top and bottom by the conjugate
of the denominator.
(3)2 – (43)2 = 9 – 48 = –39
= –39
15 – 63 – 203 + 24
= 13(3 – 23)
–39

55. b.
5 – 23
3 + 43
=
(3 – 43)
(3 – 43)
·
(5 – 23) (3 – 43)
= –39
=
39 – 263
–39
–1
More on Algebra of Radicals
(5 – 23)
(3 + 43)
Multiply the top and bottom by the conjugate
of the denominator.
(3)2 – (43)2 = 9 – 48 = –39
= –39
15 – 63 – 203 + 24
= 13(3 – 23)
–393

56. b.
5 – 23
3 + 43
=
(3 – 43)
(3 – 43)
·
(5 – 23) (3 – 43)
= –39
=
39 – 263
–39
–1
More on Algebra of Radicals
(5 – 23)
(3 + 43)
Multiply the top and bottom by the conjugate
of the denominator.
(3)2 – (43)2 = 9 – 48 = –39
= –39
15 – 63 – 203 + 24
= 13(3 – 23)
–393
= –3 + 23
3

57. Exercise A. Simplify.
1. 12 (3 + 32) 2. 8 (3 + 312)
3. 6 (43 – 52) 4. 20 (45 – 5)
5. (3 – 22)(2 + 32) 6. (5 – 23)(2 + 3)
15. (33 – 22)(23 + 32) 16. (22 –5)(42 + 35)
7. (3 – 5) (23+ 3) 8. (26 – 3) (26 + 3)
9. (43 – 2) (43 + 2) 10. (52 + 3) (52 – 3)
11. (23 – 5) (23 + 5) 12. (23 + 5) (23 + 5)
13. (43 – 2) (43 – 2) 14. (52 + 3) (52 + 3)
17. (25 – 23)(45 –5) 18. (27 –3)(47 + 33)
19. (4x – 2) (4x + 2) 20. (5x + 3) (5x – 3)
21. (4x + 2) (4x + 2) 22. (5x + 3) (5x + 3)
23. (x + h – x ) (x + h + x)
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58. Exercise B. Divide. Rationalize the denominator.
24.
1 – 3
1 + 3
25.
5 + 2
3 – 2
26.
1 – 33
2 + 3
27.
1 – 53
4 + 23
28. 32 – 33
22 – 43
29. 25 + 22
34 – 32
30. 42 – 37
22 – 27
31.
x + 3
x – 3
32. 3x – 3
3x + 2
33.
x – 2
x + 2 + 2
34.
x – 4
x – 3 – 1
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