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
= 362
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
= 362
x·x = x
3
More on Algebra of Radicals
Example A. Simplify
2
b. 12 (3 + 32)
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
= 362
x·x = x
3
More on Algebra of Radicals
Example A. Simplify
2
b. 12 (3 + 32)
= 123 + 3122
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
= 362
x·x = x
3
More on Algebra of Radicals
Example A. Simplify
2
b. 12 (3 + 32)
= 123 + 3122
= 36+ 324
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
= 362
x·x = x
3
More on Algebra of Radicals
Example A. Simplify
2
b. 12 (3 + 32)
= 123 + 3122
= 36+ 324
= 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
= 362
x·x = x
3
More on Algebra of Radicals
Example A. Simplify
2
b. 12 (3 + 32)
= 123 + 3122
= 36+ 324
= 6 + 3√4*6
= 6 + 3*26
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
= 362
x·x = x
3
More on Algebra of Radicals
Example A. Simplify
2
b. 12 (3 + 32)
= 123 + 3122
= 36+ 324
= 6 + 3√4*6
= 6 + 3*26
= 6 + 66
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
= 362
x·x = x
3
More on Algebra of Radicals
Example A. Simplify
2
b. 12 (3 + 32)
= 123 + 3122
= 36+ 324
= 6 + 3√4*6
= 6 + 3*26
= 6 + 66
a.
(Remember 6 + 6√6 = 12√6 because they are not
like-terms.)
y
x
y
x
=,
16. c. (33 – 22)(23 + 32)
More on Algebra of Radicals
17. c. (33 – 22)(23 + 32)
= 33*23
More on Algebra of Radicals
18. c. (33 – 22)(23 + 32)
= 33*23 + 33*32
More on Algebra of Radicals
19. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23
More on Algebra of Radicals
20. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
21. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
3
22. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
3 √6 √6
23. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
3 2√6 √6
24. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
= 18 + 96 – 46 – 12
3 2√6 √6
25. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
= 18 + 96 – 46 – 12
= 6 + 56
3 2√6 √6
26. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
= 18 + 96 – 46 – 12
= 6 + 56
3 2√6 √6
Conjugates
27. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
= 18 + 96 – 46 – 12
= 6 + 56
3 2√6 √6
Conjugates
We call x + y, x – y the conjugate of each other.
28. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
= 18 + 96 – 46 – 12
= 6 + 56
3 2√6 √6
Conjugates
We call x + y, x – y the conjugate of each other.
For example, the conjugate of 3 – 25 is 3 + 25,
29. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
= 18 + 96 – 46 – 12
= 6 + 56
3 2√6 √6
Conjugates
We call x + y, x – y the conjugate of each other.
For example, the conjugate of 3 – 25 is 3 + 25,
the conjugate of 5 + 22 is 5 – 22.
30. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
= 18 + 96 – 46 – 12
= 6 + 56
3 2√6 √6
Conjugates
We call x + y, x – y the conjugate of each other.
For example, the conjugate of 3 – 25 is 3 + 25,
the conjugate of 5 + 22 is 5 – 22.
The importance of conjugate pair is that
(x + y)(x – y) = x2 – y2
31. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
= 18 + 96 – 46 – 12
= 6 + 56
3 2√6 √6
Conjugates
We call x + y, x – y the conjugate of each other.
For example, the conjugate of 3 – 25 is 3 + 25,
the conjugate of 5 + 22 is 5 – 22.
The importance of conjugate pair is that
(x + y)(x – y) = x2 – y2
Example B. Multiply the following conjugates.
a. (3 – 25)(3 + 25)
32. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
= 18 + 96 – 46 – 12
= 6 + 56
3 2√6 √6
Conjugates
We call x + y, x – y the conjugate of each other.
For example, the conjugate of 3 – 25 is 3 + 25,
the conjugate of 5 + 22 is 5 – 22.
The importance of conjugate pair is that
(x + y)(x – y) = x2 – y2
Example B. Multiply the following conjugates.
a. (3 – 25)(3 + 25)
= 32 – (25)2
33. c. (33 – 22)(23 + 32)
= 33*23 + 33*32 – 22*23 – 22*32
More on Algebra of Radicals
= 18 + 96 – 46 – 12
= 6 + 56
3 2√6 √6
Conjugates
We call x + y, x – y the conjugate of each other.
For example, the conjugate of 3 – 25 is 3 + 25,
the conjugate of 5 + 22 is 5 – 22.
The importance of conjugate pair is that
(x + y)(x – y) = x2 – y2
Example B. Multiply the following conjugates.
a. (3 – 25)(3 + 25)
= 32 – (25)2
= 9 – 4*5
34. b.(5 + 22)(3 – 7)(5 – 22)(3 + 7)
More on Algebra of Radicals
35. b.(5 + 22)(3 – 7)(5 – 22)(3 + 7)
=(5 + 22)(5 – 22)(3 – 7)(3 + 7)
More on Algebra of Radicals
39. b.(5 + 22)(3 – 7)(5 – 22)(3 + 7)
=(5 + 22)(5 – 22)(3 – 7)(3 + 7)
= ((5)2 – (22)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 + 22)(3 – 7)(5 – 22)(3 + 7)
=(5 + 22)(5 – 22)(3 – 7)(3 + 7)
= ((5)2 – (22)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 – 35
a.
41. b.(5 + 22)(3 – 7)(5 – 22)(3 + 7)
=(5 + 22)(5 – 22)(3 – 7)(3 + 7)
= ((5)2 – (22)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 – 35
a.
Multiply the top and bottom by the conjugate
of the denominator.
42. b.(5 + 22)(3 – 7)(5 – 22)(3 + 7)
=(5 + 22)(5 – 22)(3 – 7)(3 + 7)
= ((5)2 – (22)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 – 35
a.
=
2
(2 – 35)
Multiply the top and bottom by the conjugate
of the denominator.
43. b.(5 + 22)(3 – 7)(5 – 22)(3 + 7)
=(5 + 22)(5 – 22)(3 – 7)(3 + 7)
= ((5)2 – (22)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 – 35
a.
=
2
(2 – 35) (2 + 35)
(2 + 35)
Multiply the top and bottom by the conjugate
of the denominator.
44. b.(5 + 22)(3 – 7)(5 – 22)(3 + 7)
=(5 + 22)(5 – 22)(3 – 7)(3 + 7)
= ((5)2 – (22)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 – 35
a.
=
2
(2 – 35) (2 + 35)
(2 + 35)
Multiply the top and bottom by the conjugate
of the denominator.
(2)2 – (35)2 = 4 – 45 = –41
45. b.(5 + 22)(3 – 7)(5 – 22)(3 + 7)
=(5 + 22)(5 – 22)(3 – 7)(3 + 7)
= ((5)2 – (22)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 – 35
a.
=
2
(2 – 35) (2 + 35)
(2 + 35)
Multiply the top and bottom by the conjugate
of the denominator.
= 4 + 6√5
– 41
(2)2 – (35)2 = 4 – 45 = –41
47. b.
5 – 23
3 + 43
More on Algebra of Radicals
Multiply the top and bottom by the conjugate
of the denominator.
48. b.
5 – 23
3 + 43
=
(3 – 43)
(3 – 43)
·
More on Algebra of Radicals
(5 – 23)
(3 + 43)
Multiply the top and bottom by the conjugate
of the denominator.
49. b.
5 – 23
3 + 43
=
(3 – 43)
(3 – 43)
·
More on Algebra of Radicals
(5 – 23)
(3 + 43)
Multiply the top and bottom by the conjugate
of the denominator.
(3)2 – (43)2
50. b.
5 – 23
3 + 43
=
(3 – 43)
(3 – 43)
·
More on Algebra of Radicals
(5 – 23)
(3 + 43)
Multiply the top and bottom by the conjugate
of the denominator.
(3)2 – (43)2 = 9 – 48 = –39
51. b.
5 – 23
3 + 43
=
(3 – 43)
(3 – 43)
·
(5 – 23) (3 – 43)
More on Algebra of Radicals
(5 – 23)
(3 + 43)
Multiply the top and bottom by the conjugate
of the denominator.
(3)2 – (43)2 = 9 – 48 = –39
= –39
52. b.
5 – 23
3 + 43
=
(3 – 43)
(3 – 43)
·
(5 – 23) (3 – 43)
= –39
More on Algebra of Radicals
(5 – 23)
(3 + 43)
Multiply the top and bottom by the conjugate
of the denominator.
(3)2 – (43)2 = 9 – 48 = –39
= –39
15 – 63 – 203 + 24
53. b.
5 – 23
3 + 43
=
(3 – 43)
(3 – 43)
·
(5 – 23) (3 – 43)
= –39
=
39 – 263
–39
More on Algebra of Radicals
(5 – 23)
(3 + 43)
Multiply the top and bottom by the conjugate
of the denominator.
(3)2 – (43)2 = 9 – 48 = –39
= –39
15 – 63 – 203 + 24
54. b.
5 – 23
3 + 43
=
(3 – 43)
(3 – 43)
·
(5 – 23) (3 – 43)
= –39
=
39 – 263
–39
More on Algebra of Radicals
(5 – 23)
(3 + 43)
Multiply the top and bottom by the conjugate
of the denominator.
(3)2 – (43)2 = 9 – 48 = –39
= –39
15 – 63 – 203 + 24
= 13(3 – 23)
–39
55. b.
5 – 23
3 + 43
=
(3 – 43)
(3 – 43)
·
(5 – 23) (3 – 43)
= –39
=
39 – 263
–39
–1
More on Algebra of Radicals
(5 – 23)
(3 + 43)
Multiply the top and bottom by the conjugate
of the denominator.
(3)2 – (43)2 = 9 – 48 = –39
= –39
15 – 63 – 203 + 24
= 13(3 – 23)
–393
56. b.
5 – 23
3 + 43
=
(3 – 43)
(3 – 43)
·
(5 – 23) (3 – 43)
= –39
=
39 – 263
–39
–1
More on Algebra of Radicals
(5 – 23)
(3 + 43)
Multiply the top and bottom by the conjugate
of the denominator.
(3)2 – (43)2 = 9 – 48 = –39
= –39
15 – 63 – 203 + 24
= 13(3 – 23)
–393
= –3 + 23
3