rules of radicals

1. Rules of Radicals

2. Rules of Radicals
All variables are assumed
to be non-negative
in the following discussion.

3. Square Rule: x2 =x x = x.
Rules of Radicals
All variables are assumed
to be non-negative
in the following discussion.

4. Square Rule: x2 =x x = x.
Multiplication Rule: x·y = x·y
Rules of Radicals
All variables are assumed
to be non-negative
in the following discussion.

5. Example A. Simplify
a. 8
Square Rule: x2 =x x = x.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
Rules of Radicals
All variables are assumed
to be non-negative
in the following discussion.

6. Example A. Simplify
a. 8 =
Square Rule: x2 =x x = x.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors 4, 9, 16, 25, 36,..
of the radicand to extract when simplifying square-root.
Rules of Radicals
All variables are assumed
to be non-negative
in the following discussion.

7. Example A. Simplify
a. 8 =
Square Rule: x2 =x x = x.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors 4, 9, 16, 25, 36,..
of the radicand to extract when simplifying square-root.
Rules of Radicals
All variables are assumed
to be non-negative
in the following discussion.
The square number 4
is a factor of 8.

8. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors 4, 9, 16, 25, 36,..
of the radicand to extract when simplifying square-root.
Rules of Radicals
All variables are assumed
to be non-negative
in the following discussion.
The square number 4
is a factor of 8.

9. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors 4, 9, 16, 25, 36,..
of the radicand to extract when simplifying square-root.
Rules of Radicals
b. 72 =
All variables are assumed
to be non-negative
in the following discussion.
The square number 36
is a factor of 72.

10. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors 4, 9, 16, 25, 36,..
of the radicand to extract when simplifying square-root.
Rules of Radicals
b. 72 =362
All variables are assumed
to be non-negative
in the following discussion.
The square number 36
is a factor of 72.

11. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors 4, 9, 16, 25, 36,..
of the radicand to extract when simplifying square-root.
Rules of Radicals
b. 72 =362 = 62
All variables are assumed
to be non-negative
in the following discussion.
The square number 36
is a factor of 72.

12. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors 4, 9, 16, 25, 36,..
of the radicand to extract when simplifying square-root.
Rules of Radicals
b. 72 =362 = 62
c. x2y
All variables are assumed
to be non-negative
in the following discussion.

13. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors 4, 9, 16, 25, 36,..
of the radicand to extract when simplifying square-root.
Rules of Radicals
b. 72 =362 = 62
c. x2y =x2y
All variables are assumed
to be non-negative
in the following discussion.

14. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors 4, 9, 16, 25, 36,..
of the radicand to extract when simplifying square-root.
Rules of Radicals
b. 72 =362 = 62
c. x2y =x2y = xy
All variables are assumed
to be non-negative
in the following discussion.

15. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors 4, 9, 16, 25, 36,..
of the radicand to extract when simplifying square-root.
Rules of Radicals
b. 72 =362 = 62
d. x2y3
c. x2y =x2y = xy
All variables are assumed
to be non-negative
in the following discussion.

16. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors 4, 9, 16, 25, 36,..
of the radicand to extract when simplifying square-root.
Rules of Radicals
b. 72 =362 = 62
d. x2y3 =x2y2y
c. x2y =x2y = xy
All variables are assumed
to be non-negative
in the following discussion.

17. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors 4, 9, 16, 25, 36,..
of the radicand to extract 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
All variables are assumed
to be non-negative
in the following discussion.

18. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors 4, 9, 16, 25, 36,..
of the radicand to extract 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
All variables are assumed
to be non-negative
in the following discussion.
A radical expression is said to be simplified
if as much as possible is extracted out of the radical.

19. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors 4, 9, 16, 25, 36,..
of the radicand to extract 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
All variables are assumed
to be non-negative
in the following discussion.
(For example 72 =98 = 38 are not simplified, but 62 is.)
A radical expression is said to be simplified
if as much as possible is extracted out of the radical.

20. Rules of Radicals
We may simplify a radical expression by extracting roots
out of the radical in steps.

21. Example B. Simplify.
a. 72
Rules of Radicals
We may simplify a radical expression by extracting roots
out of the radical in steps.

22. Example B. Simplify.
a. 72 = 4 18
Rules of Radicals
We may simplify a radical expression by extracting roots
out of the radical in steps.

23. Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
Rules of Radicals
We may simplify a radical expression by extracting roots
out of the radical in steps.

24. Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292
Rules of Radicals
We may simplify a radical expression by extracting roots
out of the radical in steps.

25. We may simplify a radical expression by extracting roots
out of the radical in steps.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2
Rules of Radicals

26. Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
Rules of Radicals
We may simplify a radical expression by extracting roots
out of the radical in steps.

27. 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
We may simplify a radical expression by extracting roots
out of the radical in steps.

28. 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
We may simplify a radical expression by extracting roots
out of the radical in steps.

29. 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
We may simplify a radical expression by extracting roots
out of the radical in steps.

30. 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
 =
We may simplify a radical expression by extracting roots
out of the radical in steps.

31. 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.
We may simplify a radical expression by extracting roots
out of the radical in steps.

32. 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.
We may simplify a radical expression by extracting roots
out of the radical in steps.

33. 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. =
We may simplify a radical expression by extracting roots
out of the radical in steps.

34. 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. = =
We may simplify a radical expression by extracting roots
out of the radical in steps.

35. 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.
We may simplify a radical expression by extracting roots
out of the radical in steps.

36. 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. =
We may simplify a radical expression by extracting roots
out of the radical in steps.

37. 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. = =
We may simplify a radical expression by extracting roots
out of the radical in steps.

38. The root of a fraction is said to be simplified
if the denominator does not have any root-terms.
Rules of Radicals
Example D. Simplify
5
3a. 

39. The root of a fraction is said to be simplified
if the denominator does not have any root-terms.
If not, the denominator needs to be changed to be root free.
Rules of Radicals
Example D. Simplify
5
3a. 

40. The root of a fraction is said to be simplified
if the denominator does not have any root-terms.
If not, the denominator needs to be changed to be root 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.
Rules of Radicals
Example D. Simplify
5
3a. 

41. Example D. Simplify
5
3
5·5
3·5
The root of a fraction is said to be simplified
if the denominator does not have any root-terms.
If not, the denominator needs to be changed to be root 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.
Rules of Radicals
a. = 
Multiple an extra 5
to extract root to make
denominator to be root-free.

42. Example D. Simplify
5
3
5·5
3·5
The root of a fraction is said to be simplified
if the denominator does not have any root-terms.
If not, the denominator needs to be changed to be root 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.
Rules of Radicals
a. =  =
25
15

Multiple an extra 5
to extract root to make
denominator to be root-free.

43. Example D. Simplify
5
3
5·5
3·5
The root of a fraction is said to be simplified
if the denominator does not have any root-terms.
If not, the denominator needs to be changed to be root 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.
Rules of Radicals
a. =  =
25
15

=
5
15
Multiple an extra 5
to extract root to make
denominator to be root-free.
Simplified! Became the
denominator is root-free
(and the numerator is simplified).

44. Example D. Simplify
5
3
5·5
3·5
The root of a fraction is said to be simplified
if the denominator does not have any root-terms.
If not, the denominator needs to be changed to be root 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.
Rules of Radicals
a. =  =
25
15

=
5
15
Multiple an extra 5
to extract root to make
denominator to be root-free.
Simplified! Became the
denominator is root-free
(and the numerator is simplified).
5
1 15or

45. Example D. Simplify
5
3
5·5
3·5
The root of a fraction is said to be simplified
if the denominator does not have any root-terms.
If not, the denominator needs to be changed to be root 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.
Rules of Radicals
a. =  =
25
15

=
5
15
8x
5b. 
5
1 15or

46. Example D. Simplify
5
3
5·5
3·5
The root of a fraction is said to be simplified
if the denominator does not have any root-terms.
If not, the denominator needs to be changed to be root 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.
Rules of Radicals
a. =  =
25
15

=
5
15
8x
5
4·2x
5b. = 
5
1 15or

47. Example D. Simplify
5
3
5·5
3·5
The root of a fraction is said to be simplified
if the denominator does not have any root-terms.
If not, the denominator needs to be changed to be root 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.
Rules of Radicals
2
a. =  =
25
15

=
5
15
8x
5
4·2x
5b. =  =
2x
5

5
1 15or

48. Example D. Simplify
5
3
5·5
3·5
The root of a fraction is said to be simplified
if the denominator does not have any root-terms.
If not, the denominator needs to be changed to be root 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.
Rules of Radicals
2
a. =  =
25
15

=
5
15
8x
5
4·2x
5b. =  =
2x
5

=
2 2x
5
 2x
2x

5
1 15or

49. Example D. Simplify
5
3
5·5
3·5
The root of a fraction is said to be simplified
if the denominator does not have any root-terms.
If not, the denominator needs to be changed to be root 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.
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

50. Example D. Simplify
5
3
5·5
3·5
The root of a fraction is said to be simplified
if the denominator does not have any root-terms.
If not, the denominator needs to be changed to be root 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.
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

51. Example D. Simplify
5
3
5·5
3·5
The root of a fraction is said to be simplified
if the denominator does not have any root-terms.
If not, the denominator needs to be changed to be root 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.
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

52. Example D. Simplify
5
3
5·5
3·5
The root of a fraction is said to be simplified
if the denominator does not have any root-terms.
If not, the denominator needs to be changed to be root 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.
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

53. Example D. Simplify
5
3
5·5
3·5
The root of a fraction is said to be simplified
if the denominator does not have any root-terms.
If not, the denominator needs to be changed to be root 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.
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 root of a fraction is said to be simplified
if the denominator does not have any root-terms.
If not, the denominator needs to be changed to be root 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.
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

55. Example D. Simplify
5
3
5·5
3·5
The root of a fraction is said to be simplified
if the denominator does not have any root-terms.
If not, the denominator needs to be changed to be root 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.
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

56. Example D. Simplify
5
3
5·5
3·5
The root of a fraction is said to be simplified
if the denominator does not have any root-terms.
If not, the denominator needs to be changed to be root 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.
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

57. 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.

58. 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.