The document discusses rules for simplifying radical expressions. It states the square root and multiplication rules, which are that the square root of a squared term is the term itself, and that the square root of a product is the product of the square roots. Examples are provided to demonstrate applying these rules to simplify radical expressions by extracting square factors from the radicand. The division rule for radicals is also stated.

1. Rules of Radicals

2. Square Rule: x2 =x x = x if x > 0.
Rules of Radicals

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 = 42
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 = 42 = 22
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 = 42 = 22
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 = 42 = 22
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 =362

11. Example A. Simplify
a. 8 = 42 = 22
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 =362 = 62

12. Example A. Simplify
a. 8 = 42 = 22
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 =362 = 62
c. x2y

13. Example A. Simplify
a. 8 = 42 = 22
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 =362 = 62
c. x2y =x2y

14. Example A. Simplify
a. 8 = 42 = 22
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 =362 = 62
c. x2y =x2y = xy

15. Example A. Simplify
a. 8 = 42 = 22
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 =362 = 62
d. x2y3
c. x2y =x2y = xy

16. Example A. Simplify
a. 8 = 42 = 22
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 =362 = 62
d. x2y3 =x2y2y
c. x2y =x2y = xy

17. Example A. Simplify
a. 8 = 42 = 22
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 =362 = 62
d. x2y3 =x2y2y = xyy
c. x2y =x2y = xy

18. Example A. Simplify
a. 8 = 42 = 22
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 =362 = 62
d. x2y3 =x2y2y = xyy
c. x2y =x2y = xy
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 = 218 (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 = 218 (not simplified yet)
= 292
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 = 218 (not simplified yet)
= 292 = 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 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (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 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (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 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (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 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
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 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
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 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
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 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
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 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
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 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
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 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
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 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
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 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
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 + 913 =
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 + 913 =
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 +913 =
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 = 513 =
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. 1818 21. 2 16
23. 183
22. 123
24. 1227 25. 1850 26. 1040
27. 20x15x 28.12xy15y
29. 32xy324x5 30. x8y13x15y9
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.