3. Square Rule: x2 =x x = x if x > 0.
Multiplication Rule: x·y = x·y
Rules of Radicals
4. Square Rule: x2 =x x = x if x > 0.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
Rules of Radicals
5. Square Rule: x2 =x x = x if x > 0.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals
6. Example A. Simplify
a. 8
Square Rule: x2 =x x = x if x > 0.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals
7. Example A. Simplify
a. 8 = 42
Square Rule: x2 =x x = x if x > 0.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals
8. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x if x > 0.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals
9. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x if x > 0.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals
b. 72 =
10. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x if x > 0.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals
b. 72 =362
11. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x if x > 0.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals
b. 72 =362 = 62
12. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x if x > 0.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals
b. 72 =362 = 62
c. x2y
13. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x if x > 0.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals
b. 72 =362 = 62
c. x2y =x2y
14. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x if x > 0.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals
b. 72 =362 = 62
c. x2y =x2y = xy
15. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x if x > 0.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals
b. 72 =362 = 62
d. x2y3
c. x2y =x2y = xy
16. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x if x > 0.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals
b. 72 =362 = 62
d. x2y3 =x2y2y
c. x2y =x2y = xy
17. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x if x > 0.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals
b. 72 =362 = 62
d. x2y3 =x2y2y = xyy
c. x2y =x2y = xy
18. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x if x > 0.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals
b. 72 =362 = 62
d. x2y3 =x2y2y = xyy
c. x2y =x2y = xy
A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
19. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Rules of Radicals
20. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72
Rules of Radicals
21. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18
Rules of Radicals
22. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
Rules of Radicals
23. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292
Rules of Radicals
24. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2
Rules of Radicals
25. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
Rules of Radicals
26. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5
Rules of Radicals
27. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
Rules of Radicals
28. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
Rules of Radicals
29. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
Rules of Radicals
Division Rule: y
x
y
x
=
30. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
Rules of Radicals
Division Rule: y
x
y
x
=
Example C. Simplify.
31. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
Rules of Radicals
Division Rule: y
x
y
x
=
Example C. Simplify.
9
4
a.
32. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
Rules of Radicals
Division Rule: y
x
y
x
=
Example C. Simplify.
9
4
9
4
a. =
33. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
Rules of Radicals
Division Rule: y
x
y
x
=
Example C. Simplify.
9
4
9
4
3
2
a. = =
34. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
Rules of Radicals
Division Rule: y
x
y
x
=
Example C. Simplify.
9
4
9
4
3
2
9y2
x2
a. = =
b.
35. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
Rules of Radicals
Division Rule: y
x
y
x
=
Example C. Simplify.
9
4
9
4
3
2
9y2
x2
9y2
x2
a. = =
b. =
36. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
Rules of Radicals
Division Rule: y
x
y
x
=
Example C. Simplify.
9
4
9
4
3
2
9y2
x2
9y2
x2
3y
x
a. = =
b. = =
37. The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
Rules of Radicals
38. The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
Rules of Radicals
39. The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
40. Example D. Simplify
5
3
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
a.
41. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
a. =
42. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
a. = =
25
15
43. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
a. = =
25
15
=
5
15
44. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
a. = =
25
15
=
5
15
8x
5b.
5
1 15or
45. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
a. = =
25
15
=
5
15
8x
5
4·2x
5b. =
5
1 15or
46. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
2
a. = =
25
15
=
5
15
8x
5
4·2x
5b. = =
2x
5
5
1 15or
47. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
2
a. = =
25
15
=
5
15
8x
5
4·2x
5b. = =
2x
5
=
2 2x
5
2x
2x
5
1 15or
48. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
2
a. = =
25
15
=
5
15
8x
5
4·2x
5b. = =
2x
5
=
2 2x
5
2x
2x
=
2 2x
10x
*
5
1 15or
49. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
2
a. = =
25
15
=
5
15
8x
5
4·2x
5b. = =
2x
5
=
2 2x
5
2x
2x
=
2 2x
10x
*
=
4x
10x
5
1 15or
50. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
2
a. = =
25
15
=
5
15
8x
5
4·2x
5b. = =
2x
5
=
2 2x
5
2x
2x
=
2 2x
10x
*
=
4x
10x
5
1 15or
4x
1 10xor
51. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
2
a. = =
25
15
=
5
15
8x
5
4·2x
5b. = =
2x
5
=
2 2x
5
2x
2x
=
2 2x
10x
*
=
4x
10x
WARNING!!!!
a ± b = a ±b
5
1 15or
4x
1 10xor
52. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
2
a. = =
25
15
=
5
15
8x
5
4·2x
5b. = =
2x
5
=
2 2x
5
2x
2x
=
2 2x
10x
*
=
4x
10x
WARNING!!!!
a ± b = a ±b
For example: 4 + 913 =
5
1 15or
4x
1 10xor
53. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
2
a. = =
25
15
=
5
15
8x
5
4·2x
5b. = =
2x
5
=
2 2x
5
2x
2x
=
2 2x
10x
*
=
4x
10x
WARNING!!!!
a ± b = a ±b
For example: 4 + 913 =
5
1 15or
4x
1 10xor
54. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
2
a. = =
25
15
=
5
15
8x
5
4·2x
5b. = =
2x
5
=
2 2x
5
2x
2x
=
2 2x
10x
*
=
4x
10x
WARNING!!!!
a ± b = a ±b
For example: 4 + 9 = 4 +913 =
5
1 15or
4x
1 10xor
55. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
2
a. = =
25
15
=
5
15
8x
5
4·2x
5b. = =
2x
5
=
2 2x
5
2x
2x
=
2 2x
10x
*
=
4x
10x
WARNING!!!!
a ± b = a ±b
For example: 4 + 9 = 4 +9 = 2 + 3 = 513 =
5
1 15or
4x
1 10xor
56. Rules of Radicals
Exercise A. Simplify the following radicals.
1. 12 2. 18 3. 20 4. 28
5. 32 6. 36 7. 40 8. 45
9. 54 10. 60 11. 72 12. 84
13. 90 14. 96x2 15. 108x3 16. 120x2y2
17. 150y4 18. 189x3y2 19. 240x5y8 18. 242x19y34
19. 12 12 20. 1818 21. 2 16
23. 183
22. 123
24. 1227 25. 1850 26. 1040
27. 20x15x 28.12xy15y
29. 32xy324x5 30. x8y13x15y9
Exercise B. Simplify the following radicals. Remember that
you have a choice to simplify each of the radicals first then
multiply, or multiply the radicals first then simplify.
57. Rules of Radicals
Exercise C. Simplify the following radicals. Remember that
you have a choice to simplify each of the radicals first then
multiply, or multiply the radicals first then simplify. Make sure
the denominators are radical–free.
8x
531. x
10
14
5x32. 7
20
5
1233. 15
8x
534. 3
2
3
32x35. 7
5
5
236. 29
x
x
(x + 1)39. x
(x + 1)
x
(x + 1)40. x(x + 1)
1
1
(x + 1)
37.
x
(x2 – 1)41. x(x + 1)
(x – 1)
x
(x + 1)38.
x21 – 1
Exercise D. Take the denominators of out of the radical.
42.
9x21 – 143.