The document discusses rules for simplifying radical expressions. It states the square root and multiplication rules, which are that the square root of a perfect square is the number itself, and that the square root of a product is the product of the individual square roots. Examples are provided to demonstrate simplifying radical expressions by extracting square roots from the radicand in steps using these rules.
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How to Become a Fashion Model – Kim HaniephKim Hanieph
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Sir Ashley Harrison Attorney: Real estate lawyers serve two primary functions in the real estate world. They either act as litigators or handle the legal aspects of real estate transactions. Real estate lawyers deal with real property and mediate real estate transactions gone sour. Typically, real estate lawyers with their own practice or in small firms focus on residential real estate transactions.
Este material tem o objetivo de estimular a reflexão sobre as formas de abordagem de segurança e dos meios de tornar este processo mais atrativo para os colaboradores.
Kurkistus älykkään markkinoinnin maailmaan - Kuinka tiedolla johdettu markkin...Annalect Finland
Tarkastelimme tiedolla johdetun markkinoinnin nykytilaa Suomessa toteuttamalla laajan kyselytutkimuksen 300 suomalaiselle yritysjohtajalle. Katson tutkimuksen tulokset raportista.
This learner's module discusses or talks about the topic Radical Expressions. It also teaches how to recognize basic radical notation. It also teaches the multiplication and division of Radical Expressions.
How to Make a Field invisible in Odoo 17Celine George
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We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
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In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
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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 = 42 = 22
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 = 42 = 22
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 = 42 = 22
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 =362
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 = 42 = 22
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 =362 = 62
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 = 42 = 22
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 =362 = 62
c. x2y
All variables are assumed
to be non-negative
in the following discussion.
13. Example A. Simplify
a. 8 = 42 = 22
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 =362 = 62
c. x2y =x2y
All variables are assumed
to be non-negative
in the following discussion.
14. Example A. Simplify
a. 8 = 42 = 22
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 =362 = 62
c. x2y =x2y = xy
All variables are assumed
to be non-negative
in the following discussion.
15. Example A. Simplify
a. 8 = 42 = 22
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 =362 = 62
d. x2y3
c. x2y =x2y = xy
All variables are assumed
to be non-negative
in the following discussion.
16. Example A. Simplify
a. 8 = 42 = 22
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 =362 = 62
d. x2y3 =x2y2y
c. x2y =x2y = xy
All variables are assumed
to be non-negative
in the following discussion.
17. Example A. Simplify
a. 8 = 42 = 22
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 =362 = 62
d. x2y3 =x2y2y = xyy
c. x2y =x2y = xy
All variables are assumed
to be non-negative
in the following discussion.
18. Example A. Simplify
a. 8 = 42 = 22
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 =362 = 62
d. x2y3 =x2y2y = xyy
c. x2y =x2y = xy
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 = 42 = 22
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 =362 = 62
d. x2y3 =x2y2y = xyy
c. x2y =x2y = xy
All variables are assumed
to be non-negative
in the following discussion.
(For example 72 =98 = 38 are not simplified, but 62 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 = 218 (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 = 218 (not simplified yet)
= 292
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 = 218 (not simplified yet)
= 292 = 2*3*2
Rules of Radicals
26. Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (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 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (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 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (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 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
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 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
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 = 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.
We may simplify a radical expression by extracting roots
out of the radical in steps.
32. 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.
We may simplify a radical expression by extracting roots
out of the radical in steps.
33. 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. =
We may simplify a radical expression by extracting roots
out of the radical in steps.
34. 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. = =
We may simplify a radical expression by extracting roots
out of the radical in steps.
35. 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.
We may simplify a radical expression by extracting roots
out of the radical in steps.
36. 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. =
We may simplify a radical expression by extracting roots
out of the radical in steps.
37. 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. = =
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 + 913 =
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 + 913 =
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 +913 =
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 = 513 =
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. 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.
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.
