Algebra of Radicals

1. Algebra of Radicals

2. Algebra of Radicals
Multiplication Rule: x·y = x·y, x·x = x

3. Algebra of Radicals
Multiplication Rule: x·y = x·y, x·x = x
Division Rule: y
x
y
x
 =

4. Algebra of Radicals
Multiplication Rule: x·y = x·y, x·x = x
Division Rule: y
x
y
x
 =
Example A. Simplify.
a. 3 * 3

5. Algebra of Radicals
Multiplication Rule: x·y = x·y, x·x = x
Division Rule: y
x
y
x
 =
Example A. Simplify.
a. 3 * 3 = 3

6. Algebra of Radicals
Multiplication Rule: x·y = x·y, x·x = x
Division Rule: y
x
y
x
 =
Example A. Simplify.
a. 3 * 3 = 3
b. 33 * 3

7. 3
Algebra of Radicals
Multiplication Rule: x·y = x·y, x·x = x
Division Rule: y
x
y
x
 =
Example A. Simplify.
a. 3 * 3 = 3
b. 33 * 3 = 3 * 3

8. 3
Algebra of Radicals
Multiplication Rule: x·y = x·y, x·x = x
Division Rule: y
x
y
x
 =
Example A. Simplify.
a. 3 * 3 = 3
b. 33 * 3 = 3 * 3 = 9

9. c. (3*3)2
3
Algebra of Radicals
Multiplication Rule: x·y = x·y, x·x = x
Division Rule: y
x
y
x
 =
Example A. Simplify.
a. 3 * 3 = 3
b. 33 * 3 = 3 * 3 = 9

10. c. (3*3)2 = 3*3 * 3 *3
3
Algebra of Radicals
Multiplication Rule: x·y = x·y, x·x = x
Division Rule: y
x
y
x
 =
Example A. Simplify.
a. 3 * 3 = 3
b. 33 * 3 = 3 * 3 = 9

11. c. (3*3)2 = 3*3 * 3 *3
3
3
Algebra of Radicals
Multiplication Rule: x·y = x·y, x·x = x
Division Rule: y
x
y
x
 =
Example A. Simplify.
a. 3 * 3 = 3
b. 33 * 3 = 3 * 3 = 9

12. c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3
3
3
Algebra of Radicals
Multiplication Rule: x·y = x·y, x·x = x
Division Rule: y
x
y
x
 =
Example A. Simplify.
a. 3 * 3 = 3
b. 33 * 3 = 3 * 3 = 9

13. c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 27
3
3
Algebra of Radicals
Multiplication Rule: x·y = x·y, x·x = x
Division Rule: y
x
y
x
 =
Example A. Simplify.
a. 3 * 3 = 3
b. 33 * 3 = 3 * 3 = 9

14. c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 27
3
3
Algebra of Radicals
Multiplication Rule: x·y = x·y, x·x = x
Division Rule: y
x
y
x
 =
Example A. Simplify.
a. 3 * 3 = 3
b. 33 * 3 = 3 * 3 = 9
d. 12 * 3

15. c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 27
3
3
Algebra of Radicals
Multiplication Rule: x·y = x·y, x·x = x
Division Rule: y
x
y
x
 =
Example A. Simplify.
a. 3 * 3 = 3
b. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36

16. c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 27
3
3
Algebra of Radicals
Multiplication Rule: x·y = x·y, x·x = x
Division Rule: y
x
y
x
 =
Example A. Simplify.
a. 3 * 3 = 3
b. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6

17. c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 27
3
3
Algebra of Radicals
Multiplication Rule: x·y = x·y, x·x = x
Division Rule: y
x
y
x
 =
Example A. Simplify.
a. 3 * 3 = 3
b. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6
e. 108 *75

18. c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 27
3
3
Algebra of Radicals
Multiplication Rule: x·y = x·y, x·x = x
Division Rule: y
x
y
x
 =
Example A. Simplify.
a. 3 * 3 = 3
b. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6
e. 108 *75 = 36*3 * 25*3

19. c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 27
3
3
Algebra of Radicals
Multiplication Rule: x·y = x·y, x·x = x
Division Rule: y
x
y
x
 =
Example A. Simplify.
a. 3 * 3 = 3
b. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6
e. 108 *75 = 36*3 * 25*3 = 63 * 53

20. c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 27
3
3
Algebra of Radicals
Multiplication Rule: x·y = x·y, x·x = x
Division Rule: y
x
y
x
 =
Example A. Simplify.
a. 3 * 3 = 3
b. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6
e. 108 *75 = 36*3 * 25*3 = 63 * 53
3

21. c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 27
3
3
Algebra of Radicals
Multiplication Rule: x·y = x·y, x·x = x
Division Rule: y
x
y
x
 =
Example A. Simplify.
a. 3 * 3 = 3
b. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6
e. 108 *75 = 36*3 * 25*3 = 63 * 53 = 30*3
3

22. c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 27
3
3
Algebra of Radicals
Multiplication Rule: x·y = x·y, x·x = x
Division Rule: y
x
y
x
 =
Example A. Simplify.
a. 3 * 3 = 3
b. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6
e. 108 *75 = 36*3 * 25*3 = 63 * 53 = 30*3 = 90
3

23. c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 27
3
3
Algebra of Radicals
Multiplication Rule: x·y = x·y, x·x = x
Division Rule: y
x
y
x
 =
Example A. Simplify.
a. 3 * 3 = 3
b. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6
e. 108 *75 = 36*3 * 25*3 = 63 * 53 = 30*3 = 90
3
f. 33 2 * 2 *2 3 2

24. c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 27
3
3
Algebra of Radicals
Multiplication Rule: x·y = x·y, x·x = x
Division Rule: y
x
y
x
 =
Example A. Simplify.
a. 3 * 3 = 3
b. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6
e. 108 *75 = 36*3 * 25*3 = 63 * 53 = 30*3 = 90
3
f. 33 2 * 2 *2 3 2
= 3 * 2 * 3 3 2 2 2

25. c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 27
3
3
Algebra of Radicals
Multiplication Rule: x·y = x·y, x·x = x
Division Rule: y
x
y
x
 =
Example A. Simplify.
a. 3 * 3 = 3
b. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6
e. 108 *75 = 36*3 * 25*3 = 63 * 53 = 30*3 = 90
3
f. 33 2 * 2 *2 3 2
= 3 * 2 * 3 3 2 2 2
3 2

26. c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 27
3
3
Algebra of Radicals
Multiplication Rule: x·y = x·y, x·x = x
Division Rule: y
x
y
x
 =
Example A. Simplify.
a. 3 * 3 = 3
b. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6
e. 108 *75 = 36*3 * 25*3 = 63 * 53 = 30*3 = 90
3
f. 33 2 * 2 *2 3 2
= 3 * 2 * 3 3 2 2 2
3 2
= 3 * 3 * 2 * 2 2

27. c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 27
3
3
Algebra of Radicals
Multiplication Rule: x·y = x·y, x·x = x
Division Rule: y
x
y
x
 =
Example A. Simplify.
a. 3 * 3 = 3
b. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6
e. 108 *75 = 36*3 * 25*3 = 63 * 53 = 30*3 = 90
3
f. 33 2 * 2 *2 3 2
= 3 * 2 * 3 3 2 2 2
3 2
= 3 * 3 * 2 * 2 2
= 362

28. Example B. Simplify.
18
x
x3
2
Algebra of Radicals
a. 

29. Example B. Simplify.
18
x
x3
2
18
x
x3
2
Algebra of Radicals
a.  = 

30. Example B. Simplify.
18
x
x3
2
18
x
x3 x2
2
Algebra of Radicals
a.  = 

31. Example B. Simplify.
18
x
x3
2
18 9
x
x3 x2
2
Algebra of Radicals
a.  = 

32. Example B. Simplify.
18
x
x3
2
18 9
x
x3 x2
2
9x2
1
Algebra of Radicals
a.  =  = 

33. Example B. Simplify.
18
x
x3
2
18 9
x
x3 x2
2
9x2
1
3x
1
Algebra of Radicals
a.  =  =  =

34. Example B. Simplify.
18
x
x3
2
18 9
x
x3 x2
2
9x2
1
3x
1
14
15
35
3
Algebra of Radicals
a.  =  =  =
b.  

35. Example B. Simplify.
18
x
x3
2
18 9
x
x3 x2
2
9x2
1
3x
1
14
15
35
3
14
15
35
3
Algebra of Radicals
a.  =  =  =
b.  
= 

36. Example B. Simplify.
18
x
x3
2
18 9
x
x3 x2
2
9x2
1
3x
1
14
15
35
3
14
15
35
3
2*7*7
3*3
Algebra of Radicals
a.  =  =  =
b.  
= 
= 

37. Example B. Simplify.
18
x
x3
2
18 9
x
x3 x2
2
9x2
1
3x
1
14
15
35
3
14
15
35
3
2*7*7
3*3
Algebra of Radicals
a.  =  =  =
b.  
= 
= 
7
3
= 2
1

38. Example B. Simplify.
18
x
x3
2
18 9
x
x3 x2
2
9x2
1
3x
1
14
15
35
3
14
15
35
3
2*7*7
3*3
Algebra of Radicals
a.  =  =  =
b.  
= 
= 
7
3
= 2*2
1*2

39. Example B. Simplify.
18
x
x3
2
18 9
x
x3 x2
2
9x2
1
3x
1
14
15
35
3
14
15
35
3
2*7*7
3*3
Algebra of Radicals
a.  =  =  =
b.  
= 
= 
7
3
= 2*2
1*2
=
7*2
32

40. Example B. Simplify.
18
x
x3
2
18 9
x
x3 x2
2
9x2
1
3x
1
14
15
35
3
14
15
35
3
2*7*7
3*3
14
32
Algebra of Radicals
a.  =  =  =
b.  
= 
= 
7
3
= 2*2
1*2
=
7*2
32
=

41. Example B. Simplify.
18
x
x3
2
18 9
x
x3 x2
2
9x2
1
3x
1
14
15
35
3
14
15
35
3
2*7*7
3*3
14
32
Algebra of Radicals
a.  =  =  =
b.  
= 
= 
7
3
= 2*2
1*2
=
7*2
32
=
We simplify higher roots in similar manner, i.e. extract as
much out of the radical as possible.

42. Algebra of Radicals
Example C. Simplify.
a. 543

43. Algebra of Radicals
Example C. Simplify.
a. 54 = √27 * 23 3

44. Algebra of Radicals
Example C. Simplify.
a. 54 = √27 * 2 = 3√23 3 3

45. Algebra of Radicals
Example C. Simplify.
a. 54 = √27 * 2 = 3√23 3 3
b. x3y53

46. Algebra of Radicals
Example C. Simplify.
a. 54 = √27 * 2 = 3√23 3 3
b. x3y5 = √x3xy3y23 3

47. Algebra of Radicals
Example C. Simplify.
a. 54 = √27 * 2 = 3√23 3 3
b. x3y5 = √x3xy3y2 = xy√xy23 33

48. Algebra of Radicals
Example C. Simplify.
a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y73
b. x3y5 = √x3xy3y2 = xy√xy23 33

49. Algebra of Radicals
Example C. Simplify.
a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y7 = √8*2x3xy6y3 3
b. x3y5 = √x3xy3y2 = xy√xy23 33

50. Algebra of Radicals
Example C. Simplify.
a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y7 = √8*2x3xy6y = 2xy2√2xy3 33
b. x3y5 = √x3xy3y2 = xy√xy23 33
Radicals of the same expression may be added and
subtracted.

51. Algebra of Radicals
Example C. Simplify.
a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y7 = √8*2x3xy6y = 2xy2√2xy3 33
b. x3y5 = √x3xy3y2 = xy√xy23 33
Radicals of the same expression may be added and
subtracted. Different radicals may not be combined.

52. Algebra of Radicals
Example C. Simplify.
a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y7 = √8*2x3xy6y = 2xy2√2xy3 33
b. x3y5 = √x3xy3y2 = xy√xy23 33
Radicals of the same expression may be added and
subtracted. Different radicals may not be combined.
Example D. Simplify.
a. 3 – 33 = –2√3

53. Algebra of Radicals
Example C. Simplify.
a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y7 = √8*2x3xy6y = 2xy2√2xy3 33
b. x3y5 = √x3xy3y2 = xy√xy23 33
Radicals of the same expression may be added and
subtracted. Different radicals may not be combined.
Example D. Simplify.
a. 3 – 33 = –2√3
b. 3 – 4√2 + 7√3 – 5√2

54. Algebra of Radicals
Example C. Simplify.
a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y7 = √8*2x3xy6y = 2xy2√2xy3 33
b. x3y5 = √x3xy3y2 = xy√xy23 33
Radicals of the same expression may be added and
subtracted. Different radicals may not be combined.
Example D. Simplify.
a. 3 – 33 = –2√3
b. 3 – 4√2 + 7√3 – 5√2
= 8√3 – 9√2

55. Algebra of Radicals
Example C. Simplify.
a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y7 = √8*2x3xy6y = 2xy2√2xy3 33
b. x3y5 = √x3xy3y2 = xy√xy23 33
Radicals of the same expression may be added and
subtracted. Different radicals may not be combined.
d. 2 + 5√6
Example D. Simplify.
a. 3 – 33 = –2√3
b. 3 – 4√2 + 7√3 – 5√2
= 8√3 – 9√2

56. Algebra of Radicals
Example C. Simplify.
a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y7 = √8*2x3xy6y = 2xy2√2xy3 33
b. x3y5 = √x3xy3y2 = xy√xy23 33
Radicals of the same expression may be added and
subtracted. Different radicals may not be combined.
d. 2 + 5√6 = 2 + 5√6 , they can’t be combined because
they are not like-terms.
Example D. Simplify.
a. 3 – 33 = –2√3
b. 3 – 4√2 + 7√3 – 5√2
= 8√3 – 9√2

57. Algebra of Radicals
Example C. Simplify.
a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y7 = √8*2x3xy6y = 2xy2√2xy3 33
b. x3y5 = √x3xy3y2 = xy√xy23 33
Radicals of the same expression may be added and
subtracted. Different radicals may not be combined.
d. 2 + 5√6 = 2 + 5√6 , they can’t be combined because
they are not like-terms.
Example D. Simplify.
a. 3 – 33 = –2√3
b. 3 – 4√2 + 7√3 – 5√2
= 8√3 – 9√2
e. 2√12 – 5√27 + √18

58. Algebra of Radicals
Example C. Simplify.
a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y7 = √8*2x3xy6y = 2xy2√2xy3 33
b. x3y5 = √x3xy3y2 = xy√xy23 33
Radicals of the same expression may be added and
subtracted. Different radicals may not be combined.
d. 2 + 5√6 = 2 + 5√6 , they can’t be combined because
they are not like-terms.
Example D. Simplify.
a. 3 – 33 = –2√3
b. 3 – 4√2 + 7√3 – 5√2
= 8√3 – 9√2
e. 2√12 – 5√27 + √18 = 2√4*3 – 5√9*3 + √9*2

59. Algebra of Radicals
Example C. Simplify.
a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y7 = √8*2x3xy6y = 2xy2√2xy3 33
b. x3y5 = √x3xy3y2 = xy√xy23 33
Radicals of the same expression may be added and
subtracted. Different radicals may not be combined.
d. 2 + 5√6 = 2 + 5√6 , they can’t be combined because
they are not like-terms.
Example D. Simplify.
a. 3 – 33 = –2√3
b. 3 – 4√2 + 7√3 – 5√2
= 8√3 – 9√2
e. 2√12 – 5√27 + √18 = 2√4*3 – 5√9*3 + √9*2
= 2*2√3 – 5*3√3 + 3√2

60. Algebra of Radicals
Example C. Simplify.
a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y7 = √8*2x3xy6y = 2xy2√2xy3 33
b. x3y5 = √x3xy3y2 = xy√xy23 33
Radicals of the same expression may be added and
subtracted. Different radicals may not be combined.
d. 2 + 5√6 = 2 + 5√6 , they can’t be combined because
they are not like-terms.
Example D. Simplify.
a. 3 – 33 = –2√3
b. 3 – 4√2 + 7√3 – 5√2
= 8√3 – 9√2
e. 2√12 – 5√27 + √18 = 2√4*3 – 5√9*3 + √9*2
= 2*2√3 – 5*3√3 + 3√2 = 4√3 – 15√3 + 3√2

61. Algebra of Radicals
Example C. Simplify.
a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y7 = √8*2x3xy6y = 2xy2√2xy3 33
b. x3y5 = √x3xy3y2 = xy√xy23 33
Radicals of the same expression may be added and
subtracted. Different radicals may not be combined.
d. 2 + 5√6 = 2 + 5√6 , they can’t be combined because
they are not like-terms.
Example D. Simplify.
a. 3 – 33 = –2√3
b. 3 – 4√2 + 7√3 – 5√2
= 8√3 – 9√2
e. 2√12 – 5√27 + √18 = 2√4*3 – 5√9*3 + √9*2
= 2*2√3 – 5*3√3 + 3√2 = 4√3 – 15√3 + 3√2 = –11√3 + 3√2

62. Algebra of Radicals
e.
2
3
 – 24

63. Algebra of Radicals
simplify each radicale.
2
3
 – 24

64. Algebra of Radicals
e.
2
3
 – 24 simplify each radical
2*2
3*2
 – 4*6=
extract denominator
extract perfect–square

65. Algebra of Radicals
e.
2
3
 – 24 simplify each radical
2*2
3*2
 – 4*6=
2*2
6
 – 26=

66. Algebra of Radicals
2
simplify each radical
2*2
3*2
 – 4*6=
2*2
6
 – 26=
2
1= 6 – 26
e.
2
3
 – 24

67. Algebra of Radicals
2
simplify each radical
2*2
3*2
 – 4*6=
2*2
6
 – 26=
2
1= 6 – 26
2
–3= 6 2
–36
or
e.
2
3
 – 24

68. Algebra of Radicals
2
simplify each radical
2*2
3*2
 – 4*6=
2*2
6
 – 26=
2
1= 6 – 26
2
–3= 6 2
–36
or
e.
2
3
 – 24
Extracting the denominator out of the radical sign enable us
to identify who can be combined with whom.

69. Algebra of Radicals
2
simplify each radical
2*2
3*2
 – 4*6=
2*2
6
 – 26=
2
1= 6 – 26
2
–3= 6 2
–36
or
e.
2
3
 – 24
Extracting the denominator out of the radical sign enable us
to identify who can be combined with whom. For instance
2
1 = 0.5

70. Algebra of Radicals
2
simplify each radical
2*2
3*2
 – 4*6=
2*2
6
 – 26=
2
1= 6 – 26
2
–3= 6 2
–36
or
e.
2
3
 – 24
Extracting the denominator out of the radical sign enable us
to identify who can be combined with whom. For instance
2
1 = 0.5 is really (half of root 2),2
1 2

71. Algebra of Radicals
2
simplify each radical
2*2
3*2
 – 4*6=
2*2
6
 – 26=
2
1= 6 – 26
2
–3= 6 2
–36
or
e.
2
3
 – 24
Extracting the denominator out of the radical sign enable us
to identify who can be combined with whom. For instance
2
1 = 0.5 is really (half of root 2), hence2
1 2
2 – = 2
1 2 .
22
1


72. Algebra of Radicals
2
simplify each radical
2*2
3*2
 – 4*6=
2*2
6
 – 26=
2
1= 6 – 26
2
–3= 6 2
–36
or
e.
2
3
 – 24
Extracting the denominator out of the radical sign enable us
to identify who can be combined with whom. For instance
2
1 = 0.5 is really (half of root 2), hence2
1 2
2 – = 2
1 2 .
22
1

simplified when there is still any radical in denominator.
Therefore a fraction is considered not

73. Algebra of Radicals
Exercise A. Simplify.
7. (2*2 )2
1. 32 * 2 2. 33 * 2 3. 33 * 2 *3
8. (2*3 )2 9. (3*3 )2
10. (–2*4 )2 11. (–4*2 )2 12. (–3*5 )2
4. 2(2 )2 5. 2(3 )2 6. 3(–3 )2
13. (–2*3 ) (3*3 ) 14. (4*2 ) 2(3 )
15. (–5*5 ) (5 ) 16. (4*2 ) (22 )
17. 22 – (3)2
19. (4*3)2 – 72
18. (2)2 + (3)2
20. (3*3 )2 – (–3 )2
21. (–5*5 )2 – (5 )2 22. (–5*5 )2 – (–5 )2

74. Algebra of Radicals
Exercise B. Simplify each radical then combine.
23. 12 – 33
Example C. Simplify.
35. 163
42. 16x5y113
41. x6y43
36. – 243 37. –323
38. 813 39. –1083 40. 1283
43. 81x9y83
24. 48 – 518
25. 427 – 512 26. 48 – 518 + 750
27. 248 – 375 + 232 – 8
28. (32)(23) – (53)(42)
29. (32)(22) – (23)(32) + (22)(23) – (23)(43)
30. (43)(33) – (43)(25) + (55)(33) – (25)(45)
31. 1/8 –2 32. 21/27 – 23 33. 62/3 – 26