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. 33 * 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. 33 * 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. 33 * 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. 33 * 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. 33 * 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. 33 * 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. 33 * 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. 33 * 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. 33 * 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. 33 * 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. 33 * 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. 33 * 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. 33 * 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. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6
e. 108 *75 = 36*3 * 25*3 = 63 * 53
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. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6
e. 108 *75 = 36*3 * 25*3 = 63 * 53
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. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6
e. 108 *75 = 36*3 * 25*3 = 63 * 53 = 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. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6
e. 108 *75 = 36*3 * 25*3 = 63 * 53 = 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. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6
e. 108 *75 = 36*3 * 25*3 = 63 * 53 = 30*3 = 90
3
f. 33 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. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6
e. 108 *75 = 36*3 * 25*3 = 63 * 53 = 30*3 = 90
3
f. 33 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. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6
e. 108 *75 = 36*3 * 25*3 = 63 * 53 = 30*3 = 90
3
f. 33 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. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6
e. 108 *75 = 36*3 * 25*3 = 63 * 53 = 30*3 = 90
3
f. 33 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. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6
e. 108 *75 = 36*3 * 25*3 = 63 * 53 = 30*3 = 90
3
f. 33 2 * 2 *2 3 2
= 3 * 2 * 3 3 2 2 2
3 2
= 3 * 3 * 2 * 2 2
= 362
30. Example B. Simplify.
18
x
x3
2
18 9
x
x3 x2
2
14
15
35
3
Algebra of Radicals
a. =
b.
=
To simplify a radical expression we may simplify the
radicands first before we extract any root.
31. 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.
=
To simplify a radical expression we may simplify the
radicands first before we extract any root.
32. 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.
=
To simplify a radical expression we may simplify the
radicands first before we extract any root.
33. 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.
=
=
To simplify a radical expression we may simplify the
radicands first before we extract any root.
34. 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
To simplify a radical expression we may simplify the
radicands first before we extract any root.
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
2*7*7
3*3
14
32
Algebra of Radicals
a. = = =
b.
=
=
7
3
= 2*2
1*2
=
7*2
32
=
To simplify a radical expression we may simplify the
radicands first before we extract any root.
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
14
32
Algebra of Radicals
a. = = =
b.
=
=
7
3
= 2*2
1*2
=
7*2
32
=
We simplify higher roots in similar manner,
i.e. extract as much out of the radical as possible.
To simplify a radical expression we may simplify the
radicands first before we extract any root.
43. Algebra of Radicals
Example C. Simplify.
a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y73
b. x3y5 = √x3xy3y2 = xy√xy23 33
44. 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
45. 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
Identical radicals may be added and subtracted.
46. 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
Identical radicals may be added and subtracted.
Different radicals may not be combined.
47. 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
Identical radicals may be added and subtracted.
Different radicals may not be combined.
Example D. Simplify.
a. 3 – 33 = –2√3
48. 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
Identical radicals may be added and subtracted.
Different radicals may not be combined.
Example D. Simplify.
a. 3 – 33 = –2√3
b. 3 – 4√2 + 7√3 – 5√2
49. 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
Identical radicals may be added and subtracted.
Different radicals may not be combined.
Example D. Simplify.
a. 3 – 33 = –2√3
b. 3 – 4√2 + 7√3 – 5√2
= 8√3 – 9√2
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
Identical radicals may be added and subtracted.
Different radicals may not be combined.
d. 2 + 5√6
Example D. Simplify.
a. 3 – 33 = –2√3
b. 3 – 4√2 + 7√3 – 5√2
= 8√3 – 9√2
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
Identical radicals 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 – 33 = –2√3
b. 3 – 4√2 + 7√3 – 5√2
= 8√3 – 9√2
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
Identical radicals 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 – 33 = –2√3
b. 3 – 4√2 + 7√3 – 5√2
= 8√3 – 9√2
e. 2√12 – 5√27 + √18
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
Identical radicals 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 – 33 = –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
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
Identical radicals 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 – 33 = –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
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
Identical radicals 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 – 33 = –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
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
Identical radicals 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 – 33 = –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
2
simplify each radical
2*2
3*2
– 4*6=
2*2
6
– 26=
2
1= 6 – 26
2
–3= 6 2
–36
or
e.
2
3
– 24
63. Algebra of Radicals
2
simplify each radical
2*2
3*2
– 4*6=
2*2
6
– 26=
2
1= 6 – 26
2
–3= 6 2
–36
or
e.
2
3
– 24
Extracting the denominator out of the radical sign enable us
to identify who can be combined with whom.
64. Algebra of Radicals
2
simplify each radical
2*2
3*2
– 4*6=
2*2
6
– 26=
2
1= 6 – 26
2
–3= 6 2
–36
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
65. Algebra of Radicals
2
simplify each radical
2*2
3*2
– 4*6=
2*2
6
– 26=
2
1= 6 – 26
2
–3= 6 2
–36
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
66. Algebra of Radicals
2
simplify each radical
2*2
3*2
– 4*6=
2*2
6
– 26=
2
1= 6 – 26
2
–3= 6 2
–36
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
67. Algebra of Radicals
2
simplify each radical
2*2
3*2
– 4*6=
2*2
6
– 26=
2
1= 6 – 26
2
–3= 6 2
–36
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